It is used to express mild dismay or frustration.
You washed that silk blouse in hot water?--egads!
Egads used to mean "ye gods" and now is more closely linked to "OMG" or "oh god"; Some people think there is only one god, so egad has evolved to express that idea. At the heart of it is the notion that god is a trickster who screws with us.
We use it to say that something severely messed up has just happened.