what causes this auditory persistence illusion? Related question: aftereffects of auditory adaptation

When the radio is turned off in the middle of a song, sometimes I can still hear the song playing very quietly for tens of seconds after the radio has been turned off. It happens for repetitive songs that I have heard many times before. A couple of friends said that they have never experienced this, so there are presumably considerable differences between individuals in susceptibility to this illusion. 

What causes this illusion?
It sounds like you are describing an earworm, which is a piece of music that repeats in your head even though it is not actually playing. This is a rather common occurrence, and matches your experience of it happening only with songs you're familiar with, and especially so for repetitive songs. What makes songs get stuck in your head is not fully understood, but research has shown that has to do both with qualities of the song (certain melodic elements, intervals, tempos and repetitions are more likely to be earworms), as well as qualities of the listener (having a musical background, obsessive-compulsive tendencies, and being in a non-focused mental state increase likelihood of experiencing earworms).