Inside every pump is some sort of pin-valve. Very much like the valve that is at the end of you CO2 tank.
Ever tap that tank valve with a screw-driver? You hear a "POP" as it releases a burst of CO2 for a second.
Thats what happens inside a pump. There is a hammer that is "cocked" by the pump. When you pull the trigger, the hammer hits the pin-valve, and released CO2 into the barrel:
Here is a typical Daisy action:
Here is a typical Sheridan action:
