They do buy a broad license that gives them the right to broadcast a huge number of songs, which is sort of like a subscription to a catalog – while that subscription is active, they can play any of the covered songs on their station. And in that contract, they agree to a fee schedule for any songs that they actually choose to air.
Then they are required to keep a record of every song they broadcast, which they report to the organization whose license they bought, along with the agreed-upon fee. That company then passes that money (minus their cut) down the chain to the individual companies (usually a record company) who own the license to that specific song. Then THAT company takes a cut, and they pay the artist(s).
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