How does this circuit light an LED simply by heating a coil?
There's not a lot to go on but I would guess that the flame conductance is connecting the finger and thumb of the person holding the lighter (via the metal of the lighter) to the base of the first transistor. If you touch the input of an audio amplifer you normally hear hum/noise through the speakers so the same sort of thing is happening here; interference pick-up on the persons body is forcing some small current into the base and turning on/off the transistors (and LED) at 50/60 Hz.

The coiled up wire on the base is just to provide more surface area to the flame to get a "better" connection.

My guess is also that if a wooden match were used (instead of the lighter) the LED probably wouldn't light.