How does licensing work for the music you play?

For terrestrial broadcast or online? You go through a handful of music industry organizations such as ASCAP and BMI. For terrestrial broadcast the costs are scaled to approximate listenership.

They just pay into ASCAP, BMI and SESAC. The royalty groups disburse on estimated airplay. Non-coms pay a reduced rate. I can go mch deeper if you want. I used to work in that field.