People yell "Geronimo" when jumping off something because a test paratrooper named Aubrey Eberhardt did it to prove his courage when he jumped out of an airplane in 1940. It became an unofficial yell of U.S. airborne troops and spread throughout U.S. culture.
In 1940, the U.S. Army was testing the idea of deploying troops using parachute drops. They decided to try a mass jump in which the testers jumped out of the plane in quick succession. The men were nervous about this. To relax, they went to see a movie the night before the jump. The movie they saw was "Geronimo," about the Apache chief. After the movie and some beer, Eberhardt claimed he wasn't nervous about the jump. The other soldiers gave him a hard time about it. He said that to prove he wasn't scared, he would yell "Geronimo" when he went out of the plane.