What happen to electricity when you turn off the lights
What turning the light off means is that you don't consume power in your home (so you don't get billed for it), so the power never gets generated at the power station, and fuel never gets burnt there to make it (which the power company then don't get billed for).

That's what 'saves' means in this context. Most people would call what a battery does 'stores' rather than 'saves'.