Why does a passenger jumping out of a rapidly moving bus fall forward with his face downwards?

The upper body continues to move even as the feet try to stop the body's forward motion. The upper body moves past the feet and legs, and the person falls over onto his hands or face.

When leaving a moving bus, the passenger is still moving at the speed of the bus, and air resistance will not slow him appreciably. The first (only) thing to diminish this velocity is the friction of the soles of his shoes when they come into contact with the ground. Unfortunately, this stops the feet before it can stop the rest of the body, so that the upper body and head are carried in an arc forward, then downward to the ground -- the body is no longer supported by the feet and legs, these being some distance behind.