More blood flow to the muscle means more nutrients delivered, which produces a stimulus for growth to occur. 
Jay Cutler
The pump is highly beneficial and for many reasons. First, the increase in blood supply to the muscles helps accelerates the removal of waste products so that I can squeeze out more reps. Second, an extreme pump enhances the potential to expand muscle fascia, allowing them to grow larger. Aside from this, it just feels amazing and helps boost my intensity to the max! 
Jim Cordova
The more blood you can force into your muscles while you’re training, the greater the anabolic signal for massive growth to occur. The Blood Volume Training program is designed to increase blood flow and nutrient delivery, crank up intensity and create an unbelievable training experience. 
Eddie Latoza
The pump is very important because it draws nutrients into the muscle and makes the muscle bigger. 
Branch Warren