When you flick the piezoelectric strip to and fro, it generates enough electricity - over 105 volts - to light the neon bulb.
The piezo film is about 0.004" thick, and it is bonded on one side to a polyester layer that is 0.0005" thick. Fanning the Flicker causes it to bend and stress the piezo film. Increasing stress creates an increasing surface charge and resulting voltage. When it reaches about 105 Volts, the bulb conducts, producing a small glow. The bulb is bipolar, so this behavior will occur at both positive and negative voltages. Because the voltage builds up very rapidly, the discharge will happen 3 to 5 times for each direction of the Flicker.
With thanks to Roger G. Gilbertson, who developed the Neon Flicker for G4GX