Why don’t they stun themselves when they discharge?
They do stun themselves, actually. Electric eels have two types of shocks they can deliver. The low level one doesn’t bother the eel much but the higher level shock affects the eel as well as whatever it is trying to shock.
The eel doesn’t feel as much of a shock as say a smaller fish that the eel feeds on would feel, partly due to the skin resistance of the eel and partly due to the fact that the charge is spread out over a longer length since the organs that make the charge make up about 80 percent of the eel’s length.