Eavesdrop
What was dropping and why the relationship to listening in?
In the olden buildings were built with wide eaves which would overhang on the roof so that rainwater could fall away from the building rather than rolling down the wall and damaging the foundation of the building. The area where the water would fall was known as the eavesdrop. This was the noun usage of the word which dates back to the 9th century.
Since the eavesdrop provided cover around the building, a person could stand in the shadow of the eave and listen to conversations going on inside the building. The eaves were always considered part of the property and under Anglo-Saxon law, you could be fined for standing under the eaves to spy.
Nevertheless, people did spy. Hung around households to gather gossip. And thus the noun became a verb that described a person who stood near the eavesdrop and listened in on the conversations within the household. Such a person came to be called an Eavesdropper.