It means you are overhearing someone talk about you. But why burning?
WAG: You’re blushing.
I always thought it was just a superstition: if your ears are burning, then someone, somewhere is talking about you (not unlike the “walking on your grave” superstition.) Only later did it become a statement that you use when you walk into a room and the conversation seems to be about you. As to where the original superstition came from… well, who knows?
Sympathetic magic.
If there is something being said that involves you, or mentions your name, then your ears wil signal you.
“Ringing” was taken already, so “burning” was made the default aural psychic indicator.
Not that it solves the burning question but my Mom always said (and I still do to this day) that if you pinch your left earlobe the person talking about you will bite their tongue.
I think the blushing is the correct answer.
According to Brewer(Phrase and Fable), it goes back to Pliny
Shakespeare, in Much ado About Nothing(iii.1), says “What fire is in mine ears?”