The dummy plug system is similar to an autopilot feature for the EVA where the dummy plug overrides manual input from the pilot. It fools the EVA into thinking it is interacting with a real pilot, even though it is actually just synchronizing with the mimicked thought patterns. Rei Ayanami's data is used as the basis for the dummy plug system. However, in order for the dummy plug to work, it must be accepted by the Eva.
Outside of combat, they are just spare bodies floating in LCL for whenever Rei needs replaced due to injury or another complication.