From Mental Floss:
In the early 1940s, the United States Army was testing the idea of parachuting from planes as a way of deploying troops. The first group to really experiment with it and begin developing paratrooper techniques was a unit of 50 men known the Parachute Test Platoon.
These guys were based out of Fort Benning, Georgia, and spent most of their summer working through grueling training sessions in the afternoon heat, wearing parachutes on their backs along with the rest of their standard gear. When training was done for the day, the troops liked to loosen up and cool down a little. Usually, most of the guys went to the air-conditioned Main Post Theatre in the evenings to see whatever movie was playing.
One night in August 1940, that movie happened to be the Paramount western, Geronimo, about the Apache chief.
Continue reading the rest of the story on Mental Floss
Source:
http://www.thatericalper.com/2012/12/31/why-do-we-scream-geronimo-when-jumping-from-things/