It is a pagan way of saying omg. It comes from the saying "ye gods", and, as everyone knows, the greek and roman gods had a way of messing with people's lives. So when you say egads, in a sense, you're saying "you fuckers," but softly.
It is used to express mild dismay or frustration.

Egad is a morphed form of 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.