When Adam Neumann pitches potential investors on his startup, WeWork Cos., he likes to rev them up with a jaunt through his company’s shared office spaces.

Before arriving, the 38-year-old chief executive typically sends staffers a directive: “Activate the space.” WeWork’s employees swarm a lounge to host an impromptu party with pizza, ice cream or margaritas.

When Mr. Neumann and his guests walk in, he often remarks how the office always seems filled with life, according to several former employees.

Fueled by showmanship, an expansive vision and the occasional shot of tequila, Mr. Neumann has propelled the New York-based office-space provider into being one of the world’s richest startups. With a valuation of more than $20 billion, or about 20 times annualized revenue, it is the fourth most valuable U.S. startup after Uber Technologies Inc., Airbnb Inc. and rocket company Space Exploration Technologies Corp., known as SpaceX. WeWork’s valuation has galloped higher in each of the past five years.

Mr. Neumann has dazzled tech investors by portraying WeWork as a Silicon Valley-style company that provides a “physical social network” for millennials. Top investors include SoftBank Group Corp. and its tech-focused Vision Fund, which added $4.4 billion in August.