I'm mostly interested in what their purpose are in Fallout 4 since I seldom play Fallout 3 or New Vegas anymore, though it would be nice to finally understand how they worked for the previous fallouts.
I remember in Fallout 3 I had no idea what they were for. In Fallout 4 I see sometimes they activate an emergency frequency which I'm not sure if that's just for authenticity or if there is a purpose behind them.
According to the Fallout 3/NV wikia:
Generally, ham radio stations are only useful in that they broadcast a radio signal and serve as a beacon to lead a player to an otherwise difficult-to-find location. Players cannot affect a ham radio apart from turning the signal on or off. Ham radios are indestructible. Since all ham radios are set to the off position by default, they can be used as "breadcrumbs" for the player, turning them on to mark where he or she has been.
Only two ham radios have a more significant function in the game.
One is set up as part of the public-address system in the Super-Duper Mart. Raiders using the store as a base use the PA system to alert others in the building.
In the Operation: Anchorage add-on, Chinese technicians use ham radios to call for backup.
As for Fallout 4, it doesn't look like they serve any important purpose (see here and here)
In the RR Headquarters, one of the RR operatives is decoding the Morse Code coming over the Ham Radio on her desk.
