Originally this word had nothing to do with snooping.

Eavesdrop started off literally: first it referred to the water that fell from the eaves of a house, then it came to mean the ground where that water fell.

Eventually, eavesdropper described someone who stood within the eavesdrop of a house to overhear a conversation inside.

Over time, the word obtained its current meaning: "to listen secretly to what is said in private."