The pump is when your muscles swell up during your workout, which is caused from the excessive amount of blood going into the muscle and filling it up the same way you would fill up a water balloon. Your muscles get a very full, tight feeling and your skin becomes tighter because of this.
Why is it needed?
To achieve maximum muscular growth a pump (scientific name, hyperemia) is essential, and the only way this can be achieved is to train correctly with the right energy intake, to allow sufficient blood flow to the working muscles. Working muscles need blood to supply them with oxygen and nutrients, and remove waste products (namely, lactic acid and carbon dioxide).
When a muscle is trained, blood flow is diverted from many other bodily processes, to supply this muscle with what it needs to perform maximally. The blood first needs to become oxygenated (which is done through gaseous exchange in the alveoli of the lungs) before it is pumped to the working muscles, where it is pooled, thus resulting in the tight feeling we call the pump.
It is thought that during training, a muscle can receive up to four times the amount of blood it would ordinarily get. Why exactly do the muscles need all this blood? As mentioned, the muscles require sufficient oxygen and nutrients to continue the sustained contracting that results in a pump.
Muscle growth will also result from the fascial stretching that occurs when the muscle is pumped beyond its normal size. When this fascial layer (which can be found between the is stretched, room for continued muscle growth is made available.
Over time, the pump will also create a greater number of capillaries (tiny blood vessels),which will, in turn, provide the muscles with more nutrients and oxygen and allow for larger pumps and more growth in the long term.