TPM
When you pitch for several innings or throw several pitches, the next day you can have some soreness. Running or any other cardio that increases heart rate can help with the soreness. When a pitcher is sore, the cardio feels like you are flushing the crap (soreness) out of your body. After the cardio, you actually feel less sore and much better. That is why it is referred to as a flush run.
Pitch -> get sore
Cardio -> Feel better
The proof comes from experience.