Think about pitching as a funnel. Energy/momentum is generated by the various parts of the body...the legs, the arms, and the torso. All this energy is to be funneled into the ball. Additionally, you want all the energy to be used to propel the ball straight to the target.
Suppose a girl does the walk through drill and is moving forward or bringing her right leg (for a rightie) around. What does that mean? It means than not all of the energy was transferred to the ball during the pitch...there is some "left over". By requiring the pitcher to stand on one foot after the throw, the pitcher can tell *ON HER OWN* whether she transferred all the energy to the ball.
Suppose a girl does the walk through drill and falls to the right (for a rightie) and is "catching" herself with her right foot? That means that the momentum she generated is going at angle to the plate, meaning that she isn't throwing the ball as fast as she can.
When pitching, you will see some good pitchers "close" after release (Yukiko Ueno is a prime example). The close is done after all the momentum is transferred to the ball. (Slow motion videos of this have been a god-send.)
The question then comes up: Baseball pitchers rotate and fall forward after the throw. Shouldn't softball pitchers do the same thing?
First, go watch a video of baseball pitchers and study exactly what happens. You see the same basic pattern a softball pitcher uses: Generation of energy with the body, and then a "stop" of the pitcher at release.
After release, the baseball pitcher moves. Why? The release of a pitched baseball ball is in front of, above and two feet or more to the right (for a rightie) of the center of gravity of the pitcher. This creates all kinds of torque on the body, pulling the body to the left (for a rightie). And, of course, since the pitcher is throwing down hill as well, he will fall forward. (This isn't rocket science...just watch a baseball pitching video.)
In softball pitching, the release is below the center of gravity of the pitcher, and almost dead center (on the X and Y axis) with the center of gravity of the pitcher. The release point is inches, not feet, from the center of gravity. If thrown correctly, the forces are moving the pitching toward the plate, not to the left or right. There are no forces requiring the body to twist.