It's an odd arrangement of astable multivibrator.
As Q2 turns on, Q1 turns off. That causes a "flyback" effect in the inductor and passes current through the LED (at a higher voltage than the battery).
Q1 turning off also add positive feedback to Q2 turning on, via C1.
Once the inductor demagnetises, the voltage at the junction of that and the LED drops again - and that reduction is passed via C1 to the base of Q2, turning it off and turning Q1 on, which continues to force the inductor output towards 0V, again providing positive feedback via C1.
R2 sets the delay before Q2 starts to turn on again.
And so on, repeating.