Acres of cargo areas surround J.F.K. Airport. Rows of semi trailers attach to blocky buildings like litters of milking puppies. Zones are lettered, buildings numbered. The setting is gray and asphalt.

If you follow signs to Cargo Area D and proceed down a nondescript road near a U.S. Post Office building, however, you’ll arrive at the most colorful place for miles, a prekindergarten and day-care center with meticulous bulletin boards and kid-size activities in every nook and cranny.

JFKids Port represents the institutions upon which Mayor Bill de Blasio built his campaign for universal prekindergarten. It is a half-day facility, vying to become full-day under the mayor’s signature proposal. If he high-fived the 32 current half-day 4- and 5-year-olds who grace these halls, the gesture would be appropriate. Children like these, most of whom do not have access to full-day prekindergarten, have played a significant role in his campaign and short time in office.

In turn, the students who will sit in these tiny seats this fall will almost certainly have full-day pre-K. In just over a year, Mr. de Blasio has drastically amped up the prekindergarten conversation from a low buzz about the benefits of universal full-day (and the budgetary constraints that made it unlikely) to a fervent acceptance that New York City will roll out universal full-day pre-K starting this fall.

It hasn’t been an easy journey for the new mayor. Mr. de Blasio stumbled in the messaging war, letting his crusade at times face more questions about its tax-the-rich funding stream and a protracted battle with the governor than the substance of pre-K itself. But with this year’s state budget, finalized last weekend, Mr. de Blasio finally has the resources to implement his vision.

The issue has taken on a national scope. President Barack Obama renewed his call for expanded pre-K programs in his State of the Union speech last January. “I’m going to pull together a coalition of elected officials, business leaders and philanthropists willing to help more kids access the high-quality pre-K they need,” Mr. Obama told congressional leaders and the American public.

How did this happen? How did an underdog candidate take a vision for an expensive dream, capture the mayor’s office and catapult that dream into a progressive “of course” throughout the city and state?

The mayor himself puts it in simple terms, and the way he frames it has that sheen of inevitability that makes it hard to recall exactly why this has been so difficult. “Universal pre-K has been promised to New Yorkers for almost two decades,” Mr. de Blasio told the Observer. “We pledged the people of this city we would finally get this done for them.”

Back to JFKids Port. The sudden acceptance that universal pre-K is not just a goal but a right hasn’t come without a state-level ideological battle about funding and political credit. But politics is for grown-ups. This place is for kids, and it aims to educate more of them, all day.

JFKids Port, which serves the community around Jamaica, Queens, responded to the education department’s request for proposals to expand full-day prekindergarten access for the coming school year. School leaders hope to convert their two half-day sessions into two full-day classes. An extra classroom, currently vacant, is fully equipped down to the dress-up clothing hanging in a corner.

Parents have been asking Dawn Williams, the school’s assistant director, what will happen next year. Most of them, many of whom work, would have preferred full-day service, but the nearby public school program always fills up quickly. Of course, these same parents also vote. So do their relatives, friends, neighbors and co-workers. Mr. de Blasio, a former campaign strategist who once guided Hillary Clinton’s U.S. Senate campaign, undoubtedly has been very aware of this.

A SUDDEN GROUNDSWELL OF POPULAR support doesn’t just “happen.” It takes organization and political muscle, and that’s terrain the mayor knows well. In late December, several de Blasio campaign insiders, including his former campaign manager, Bill Hyers, and Dan Levitan of BerlinRosen, who handled press for Mr. de Blasio’s campaign, came together with several campaign donors to found UPKNYC, a nonprofit aimed at enlivening public support and organizing grassroots efforts. Union organizer Josh Gold became its campaign director, and vocal supporters included Cynthia Nixon, Olivia Wilde and Rev. Al Sharpton. Mr. Gold told the Observer: “The idea was to galvanize support among both the grassroots and influencers. … By the end of this, we signed up about 25,000 supporters. That groundwork was already there [in terms of research supporting the individual and public benefits of pre-K]. … It was just making sure that people understood that we could actually have this.”

So UPKNYC formed the outlet for pro-pre-K sentiment once the mayor was elected, but the issue predates the mayoral race for Mr. de Blasio. When Mr. de Blasio was public advocate, he focused heavily on early childhood. For several years in that office, Mr. de Blasio fought budget cuts to early childhood education, said Ursulina Ramirez, who served the public advocate’s office as a senior policy adviser for education, social welfare and service delivery and is now chief of staff to Schools Chancellor Carmen Fariña.

In advance of Mr. de Blasio’s campaign, Ms. Ramirez said, “When the opportunity came to really think about how we’re going to impact New York City’s education system, there was no doubt in our mind that early childhood education was the route to go.” It was a conclusion heavily supported by local education advocates, parents and research.

And after the fact, by polling. According to Mr. de Blasio’s pollster, Anna Greenberg of Greenberg Quinlan Rosner Research, polling on universal pre-K public acceptance “was somewhere in the 70s, and no, it didn’t change [throughout the campaign].” Ms. Greenberg, a devoted Yankees fan, has helped Chicago Mayor Rahm Emmanuel, ex-Congresswoman Gabrielle Giffords and Minnesota Gov. Mark Dayton steer their way to office, and this pollster is now having a nice New York moment.

Ms. Greenberg says that, although the issue always polled well, Mr. de Blasio’s commitment to it transcended that fact.

“First of all, the mayor came to run for mayor with the explicit goal of inequality in New York City being the driving force in the campaign. And so, it’s within that context that the universal prekindergarten issue became so important, as an indicator of his values … but also because of the role universal prekindergarten can play in reducing inequality,” she said.

An early believer in the campaign, she explained that pre-K “started off as a priority from the beginning, day one. … This is one where it really wasn’t poll-driven. It’s a happy coincidence that it’s also incredibly popular.” She added about the mayor, “I’ve never seen a poll result make him back down from any position that he holds.”

“The mayor, the public advocate at the time, really keyed in on return on investment, said Emma Wolfe, Mr. de Blasio’s former chief of staff when he was public advocate and the current director of intergovernmental affairs. From the outset of campaign conversations, prekindergarten was a main focus, she said.

“We were looking to lay down a bold and progressive message for the city. … We were looking for a policy that would touch everybody,” said Wiley Norvell, Mr. de Blasio’s deputy press secretary, who served as press secretary in the public advocate’s office. “The early education piece really started to leap off the page.”

At the same time, if this was to be the mayor’s signature issue, it had to be airtight. “He is not one for loosey goosey, ad hoc policy formation on the fly,” Ms. Wolfe said. He needed something real, not something philosophical. That’s where his tax on the rich came into play, she argued. Ms. Wolfe cannot recall a time when they discussed the issue separate from this funding stream—it was a necessity from the very beginning. So they proposed taxing the city’s high-income earners of $500,000 or more an extra 0.5 percent.

“We knew that it was going to be a big deal, and we knew we had to figure out a responsible way of paying for it,” Ms. Wolfe said. “The tax was the best vehicle that we could come up with and that the mayor could come up with to make sure it was going to be fair and reliable funding over a sustained period of time.

According to Mr. Gold of UPKNYC, prior to Mr. de Blasio, “We had yo-yo funding, basically. … I think people understood that there needed to be an effort to make sure that this program was fully funded … and make sure the elected [officials] heard that as well.”

Ms. Wolfe brushed off the idea that the tax is new; up until recently, Mr. de Blasio refused to discuss any alternative funding scenarios, insisting he would not negotiate against himself. But at the end of the day, money is money.

RECALL, IF YOU WILL, the famous “Dante ad” that made Mr. de Blasio’s son’s hairdo famous (Mr. Obama complimented him) and helped catapult his father’s campaign. Dante beseeched voters: “I want to tell you a little bit about Bill de Blasio. He’s the only Democrat with the guts to really break from the Bloomberg years, the only one who will raise taxes on the rich to fund early childhood and after-school programs.” And he goes on to list other issues.

This turning-point ad—the personal nature of it, the glorious afro, the display of the de Blasios as everyday New Yorkers—was the brainchild of John Del Cecato, a campaign consultant who signed on with Mr. de Blasio when most New Yorkers would have shrugged, “Who’s that?”

“It was a foregone conclusion that he would be making education a central piece,” Mr. Del Cecato said. Mr. de Blasio has two children, both educated in public schools. He understands politics from a parent’s point of view, the mayor himself often points out.

“The mayor is really riding a crest of knowledge and research findings. … Every stage of life depends on good, solid investments in the earlier stages, and it’s very hard to catch up when a child falls behind,” said Jeffrey Sachs, who co-founded the New York City Advisory Council on Child Well-Being. “Study after study has shown … the best investment of society, period, is in the education of its children.”

Mr. Sachs is a professor of economics at Columbia University and the director of the Earth Institute; he co-founded this council to coordinate the efforts of New York City education experts focused on this issue.

“It makes sense,” Mr. Sachs said. “It’s efficient, it’s good for the economy, it’s good for the fairness of society. … The mayor has really struck a chord.”

At the start of the campaign, Mr. de Blasio made it very clear to Mr. Del Cecato that to understand him he had to understand movement politics. “I came to learn more and more about his history of working on progressive causes and not doing so from an ivory tower but taking keys from people on the ground,” Mr. Del Cecato recalled.

Mr. Del Cecato asked the candidate to outline the issues that he felt were most important—prekindergarten was one of them. It was perfect, Mr. Del Cecato said, “because it was how we could tackle the issue of income equality in a very tangible way.” In those early days, however, Mr. Del Cecato wondered, “How do we talk about it?”

Mr. de Blasio made it clear: no politically safe pilot projects, full-day universal prekindergarten and taxes on the rich to fund it.

“He was willing to take the arrows that came along with [the tax proposal],” Mr. Del Cecato said. “I liked it, because it wasn’t the same old politician saying, [voice exaggerated like a political advertisement] ‘I’m going to do more for your schools and help our kids.’ … It was putting some meat on the bones.”

The issue was politically risky with the tax-raising bit, but then again, Mr. de Blasio was a long-shot candidate. Playing it safe would have meant continued obscurity.

“The ‘go big or go home’ approach to tackling an issue that matters to so many millions of people was really attractive to me,” Mr. Del Cecato said. He recalls Mr. de Blasio being “comfortable in his own skin” and making it very clear: “I’m not going to be something other than what I am.”

Mr. Del Cecato wasn’t sure whether it would be a successful strategy but liked that it was rooted in principle. “I said, ‘I feel like we have a shot here. And at least you’re going into it with your eyes open,’” Mr. Del Cecato said.

In October of 2012, Public Advocate de Blasio took the opportunity to spell out this ambitious education vision at an Association for a Better New York (ABNY) event. “Let’s face it,” he told the crowd of business leaders, “we’re in a city with deeply entrenched educational disparity.”

He explained that his plan for truly universal prekindergarten—full-day classes for all 4-year-old New Yorkers—is not only about educational ideals but also about economics. Prekindergarten is an investment with a return in a better-educated work force and a more productive city. He also called for expanded after-school programs for middle school students. He cautioned, “Now this is an investment; it costs money,” and went on to explain to the city’s prominent businesspeople that he would impose a five-year surcharge on those making more than $500,000 annually.

“It was a pretty quiet room,” Mr. Norvell said of the crowd’s reaction.

One year later, Mr. de Blasio returned to ABNY with the additional title of “mayoral nominee” and reiterated the same plan for “a quality prekindergarten education” and the necessity to back up these initiatives with “real dollars” from the city’s wealthiest individuals. This time, he received more enthusiastic applause.

“I was so anxious to know how things were going to poll, but at the same time, there were no illusions about him changing his stripes,” Mr. Del Cecato said. The prekindergarten push would be either working or not, but the course could not be altered.

Thankfully for Mr. Del Cecato, polls and surveys showed that the issue resonated and not just acceptably so but with eye-bulging, wowza public enthusiasm. Nearly 80 percent of voters favored the idea of prekindergarten and after-school programs supported by a tax on the rich, Mr. Del Cecato said.

“I just think we are literally, thankfully, in a place now where there’s no question about the impact of pre-K and there’s no question that this is something New Yorkers want and deserve,” Ms. Wolfe said.

MR. DE BLASIO’S PREKINDERGARTEN plan called for $340 million annually from his tax on the rich for truly universal, full-day prekindergarten. The plan estimated that 73,250 families will need full-day prekindergarten at a cost of about $10,239 per child, according to his “Ready to Launch’” implementation plan.

Currently, fewer than 27 percent of 4-year-olds can access this service, according to the mayor’s office. The others are either in half-day programs, receive full-day care through the Administration for Childhood Services and Head Start programs or do not attend school. In some high-need communities, including the area around JFKids Port, the gap between students and available full-day seats exceeds 1,000.

The catch was that the increased income tax had to be approved by the state legislature, and Gov. Andrew Cuomo and Republican lawmakers weren’t onboard. Mr. Cuomo came up with a proposal of his own—a tax-free statewide plan not limited to the five boroughs. Albany lawmakers hammered out their differences over the weekend and came up with a new number: $300 million for the city’s pre-K program—and no tax. In a statement, Mr. de Blasio hailed the deal. “With the investment announced today, this state has made a powerful and historic decision that will change the lives of tens of thousands of children. We set out down this road nearly 18 months ago. Through ups and downs, we never wavered from our promise to the people of this city to expand full-day pre-K and after-school for our children starting this September,” the mayor said Saturday. “Today that pledge became a reality.”

Mr. de Blasio’s plan, including the tax inclusion that raised it from a philosophical to a practical level, successfully pressured state lawmakers to either help him accomplish this agenda or find a viable alternative, de Blasio supporters argue. While the governor has called for universal prekindergarten in the past and expanded it statewide by 5,500 seats last year, it is doubtful that state lawmakers, regardless of whether they’re for or against the tax, would focus so heavily on this issue if it weren’t for Mr. de Blasio.

“[The tax proposal] got folks to sit up straight and showed how serious the mayor was about this policy,” said Ms. Wolfe.

Throughout recent decades, universal prekindergarten has been gaining support around the nation. Child care is a huge expense for families, especially for single-parent households and low-income families with two working parents. Though public support for full-day prekindergarten exists, often funding does not, which is why the conversation ebbs and flows with economic cycles, said Michael Griffith, the senior school finance analyst for the Education Commission of the States.

In New York State, “universal prekindergarten” was enacted in 1997, with the idea that it would gradually ramp up. The program grew to provide $385 million for statewide half-day prekindergarten programs. It never expanded to provide truly universal full-day prekindergarten.

In most states, early childhood learning programs differ from district to district, depending on budgets and priorities. While K-12 programs are covered, funding for early learning programs often relies on a combination of state funds, federal funds, private funds, specialized grants and any other available dollars, Mr. Griffith said.

According to a source close to City Hall, the argument about the tax was critical to getting this done, even now that there’s not going to be a tax. “Without the tax, we wouldn’t have gotten the money. And it was widely recognized by folks at City Hall and around City Hall, advisers to the mayor, that we needed to push for that as long as possible to make sure we got the money that we needed to. None of us believe that would have happened if we didn’t push the tax for so long.”

Given the patchwork nature, it seems strange that the state-level debate over Mr. de Blasio’s tax has centered not on the merits of the tax but on equality of education. Largely, prekindergarten is a local issue. At a state level, in most cases, it is inherently unequal. In a perfect world, it wouldn’t be, but budgets are limited.

ON A RECENT MORNING, a group of visitors including Sophia Pappas, the executive director of the Office of Early Childhood Education for the city’s Department of Education, entered a half-day prekindergarten class at JFKids Port while about 20 students poured cereal for themselves. One little boy announced proudly, “I ate all my food!”

After breakfast, teaching assistants began setting up clay and sticks for students to create little people as they learn about bones. The classroom was divided into different learning areas to encourage exploration. Activities like a Lego table (to improve fine motor skills) had been strategically placed under the recommendation of a Department of Education specialist.

The school’s proposal for next fall is for 36 full-day seats, but it has the space for more if the Department of Education determines there is a need. JFKids Port is one of about 930 community-based organizations and public schools to submit a proposal for the rollout of universal prekindergarten. With a focus on quality, Ms. Pappas’ office has been reviewing applications and expanding systems for review, instruction, teacher guidance and family support.

Ms. Pappas worked as a Teach For America prekindergarten teacher in Newark and wrote about her experience in her book, Good Morning, Children: My First Year in Early Childhood Education. She worked to develop Teach for America’s early childhood programs and studied public policy at Harvard University. For her, after three years with the education department, the city’s newfound focus on early childhood education is a dream.

“I’ve seen it, and I’ve lived it, but now being in New York, I get to actually figure out how we make this a reality in our country’s largest school district,” Ms. Pappas said.

The Department of Education, even before funding was approved, approached the expansion of universal prekindergarten under Mr. de Blasio’s plan as inevitable. The assumption is that the funding would exist.

How did we get here? In the world of politics, where change can be glacial and maddeningly incremental, this is a change that seemed to reach its tipping point on Nov. 5, 2013, the night Mayor de Blasio was elected in a landslide. The mayor had presented his vision of universal pre-K as a no-brainer. He told the Observer, “I think our vision has resonated, because it’s one that unites us all and speaks to every family’s hopes about their own children.”

Discussing the implementation starting this fall, Ms. Williams, the JFKids Port assistant director, sat next to Ms. Pappas at a kid-size table in the vacant classroom. While she still cannot officially answer parents’ questions about full-day prekindergarten at her school, it feels like official approval is merely a formality.