In "Contents of the Dead Man's Pockets," why does Tom permit himself to look down?

This is a very good question. Tom realized that his biggest problem would be to avoid looking down, because the sight could cause vertigo and make him lose his balance, in which case he could fall backwards into space. He manages to resist the understandable temptation to look down until he gets his fingers on the yellow sheet of paper. The following brief paragraph tells what happens then.

He couldn't quite touch it, and his knees now were pressed against the wall; he could bend them no farther. But by ducking his head another inch lower, the top of his head now pressed against the bricks, he lowered his right shoulder and his fingers had the paper by the corner, pulling it loose. At the same instant he saw, between his legs and far below, Lexington Avenue stretched out for miles ahead.

Apparently he only intends to look at the paper. It would seem that in that awful position and with only his fingertips touching the paper, he would have to open his eyes to see what he was doing. Otherwise, there was a risk that, after all his effort, he might pull the paper loose and it would fly away. Tom had thought that getting to the paper was the challenge, but it turns out that, because of the narrowness of the ledge, the biggest challenge is just picking the paper up. He can't look at the paper without seeing New York City all lighted up in a fascinating and terrifying panorama far below.

He saw, in that instant, the Loew's theater sign, blocks ahead past Fiftieth Street; the miles of traffic signals, all green now; the lights of cars and street lamps; countless neon signs; and the moving black dots of people. And a violent instantaneous explosion of absolute terror roared through him.

This is the high point of the story. The author has deliberately withheld any description of the city from a dizzying height until right now, when the viewpoint character is in his most awkward position on the ledge and is barely maintaining his balance.

Tom had to look down. He could close his eyes again, but he couldn't shut out the awesome vision he had seen. The "absolute terror" he felt was both a threat and a motivating factor. This terror could make him fall, but it also forces him to act. After a long inner struggle, he manages to shut out the spectacle he had seen.

With fear-soaked slowness, he slid his left foot an inch or two toward his own impossibly distant window.