Distance running does a few of things for a pitcher.
The running gets the blood pumping and moving to help relieve soreness in the arm and body. When a pitcher feels less sore he is able to get back into action throwing again. It serves as a way to recover quicker, feel better and could help avoid later injuries.
It also provides an opportunity to build endurance. A pitcher that is in shape can pitch longer in long innings, long games, and long seasons. Running can help keep pitchers physically fit to perform.
Last but not least, when you are on a distance run, you have a lot of time to think and a lot of time to listen to your inner dialogue. That voice inside your head that says keep going or quit, Ive had enough. Its the same voice you listen to when your on the mound. So in your distance running, push yourself and learn how to make yourself go when the going gets tough.
Knowing the purpose of distance running and then getting it done will give you confidence on the mound.