Originally the Supreme Court decided that they didn't (just as someone making a recording of a song didn't have to pay the publisher of the sheet music).
But the musicians unions were losing out on radio gigs when stations played records. Stations still pay into a fund to give live musicians paying gigs.
There is a radio station in my area that plays mostly pre-rock oriented music from the 1940s and early 50s, with occasional 50s and 60s oldies thrown in.
The station relies (relied) heavily on listener contributions, and the station owner would periodically go on the air begging for donations.
One of the things he talked about in the last few years was that there were large increases in the royalty payments that the station was required to pay.