Times Square, the famed Manhattan crossroads, is where Americans often mark moments of great change, and so, too, as Wednesday dawned. But instead of its usual buzz and bustle, it was filled with an eerie silence.

The costumed cartoon characters melted away. There was no music or sound from the news broadcast on the giant screens. And thousands of people, their faces illuminated by Times Square’s flashing advertisements, stood watching and waiting as Donald J. Trump steamrollered toward victory over Hillary Clinton.

“Interesting evening, isn’t it?” said Graham Simpson, an Englishman who had come to New York last weekend to run a marathon. “The world seems to have turned on its head.”

Since arriving in the United States in July to cover this tumultuous election from a foreigner’s perspective, I have sought to understand voters’ motivations — particularly when it came to Mr. Trump, an unorthodox outsider who had upended the political system. After traveling the country, I spent the final day of a surreal journey in New York City, that anomalous financial and cultural capital that is home to Mr. Trump.