Why can you 'hear' a song that isn't actually playing?

Because your brain can generate a real, structured copy of a sound without any sound actually arriving at your ears. This is called auditory imagery, a form of mental imagery used to organize and analyze sounds when there's no external auditory stimulus present, and it comes in a couple of flavours, like imagining words (verbal imagery) or imagining music (musical imagery).

It isn't a vague impression. Researchers have found that the auditory images people generate are produced in real time and carry fairly precise information about pitch, melody and harmony, close cousins of the real thing rather than a blurry memory of it. One clever experiment shows it: ask someone to sing 'Jingle Bells' silently in their head, then ask them to judge whether the word 'Snow' is higher or lower in pitch than the word 'Sleigh'. People take longer to make that judgment when the two words are further apart in the song, exactly as you'd expect if they were mentally running through the tempo of the actual melody rather than just recalling facts about it. Musicians, who have more flexible internal tempo representations, show even more control over this. Pitch tends to survive clearly in these mental replays too, but loudness usually doesn't; people can imagine a song's melody vividly while losing almost all sense of how loud any part of it was.

The flip side of controlling this ability is losing control of it. An earworm, sometimes called a brainworm or stuck song syndrome, is a catchy or memorable piece of music that keeps occupying your mind even after it has stopped playing, a particularly stubborn and involuntary form of auditory imagery. According to one researcher, James Kellaris, 98 percent of people experience earworms at some point. Most earworms come from songs with lyrics: in his figures, roughly three-quarters of them, against less than a tenth for instrumental music. A typical earworm lasts just 15 to 30 seconds, and earworms are more common in people with an interest in music.