A dictionary definition of "climax" is: The most exciting and important part of a story, play, or movie that occurs usually at or near the end; the most interesting and exciting part of something: the high point.
Tom Benecke has decided he can retrieve his important piece of paper safely if he just doesn't look down. He shuffles along the narrow ledge, hugging the wall and holding onto whatever finger-holds he can find between the bricks. He keeps the left side of his face pressed against the wall and avoids the temptation to look down at the street eleven floors below, because he knows the sight could not only make him dizzy but cause him to lose his nerve.
But he has made several miscalculations. He didn't realize the ledge was so narrow until he was actually standing on it. He has to walk on the balls of his feet because the ledge isn't even as wide as his shoes. He didn't realize it would be so dark once he got away from his window. He didn't realize it would be so cold and windy. Worst of all, he didn't realize that when he got to the paper he was going to have a hard time picking it up. This is because he cannot turn sideways on the narrow ledge and bend over at the waist. He has to keep hugging the wall and trying to lower his body enough so that he can reach the paper with his arm lowered alongside his leg. But bending his knees against the wall nearly causes him to lose his precarious balance and topple over backwards into space. He finally has to look down in order to get his fingertips on the paper. And when he does this he gets his first glimpse of the city from that awful height. Up to this point the author has cleverly avoided any description of the scene down below. This is the climax.
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 explosion of absolute terror roared through him. For a motionless instant he saw himself externally--bent practically double, balanced on this narrow ledge, nearly half his body projecting out over the street far below--and he began to tremble violently, panic flaring through his mind and muscles, and he felt the blood rush from the surface of his skin.
This paragraph really synthesizes the essence of the story. A man is standing on a ledge paralyzed with fear and dizzied by the lights and movements of the stupendous city of New York. There are some nice touches in Finney's description. The traffic signals are "all green now," but in a minute they will all turn red at the same time. The neon signs of the day were constantly changing colors, and many featured moving lights along the edges. The headlights of all the cars and the black dots of pedestrians create more motion in the scene below. All of these elements have a dizzying and unnerving effect on the lone man clinging to the brick wall and expecting to fall to his death.
The reader also expects Tom to fall with a scream, partly because the story's title, "Contents of the Dead Man's Pocket," seems to predict that fate. The reader can sense how difficult it would be to force himself to move back along that ledge to get to his window, when he was so cold and frightened that his body wouldn't obey his brain. Tom can't even trust his brain because he realizes he must be crazy to have gotten himself into this suicidal situation in the first place.