It's a simple blocking oscillator. When power is first turned on, the two transistors start conducting and the current through the coil increases, while this is happening the top is positive and through the feedback capacitor keeps the transistors on. Once it saturates, the potential difference across it is no longer enough to keep them on, and that results in the transistors turning off, causing the field in the inductor to collapse and pull current through the LED (it is now in series with the inductor and the power source, so you get a boost in voltage). Since its top is now negative, it keeps the transistors off. The transistors start turning on again once the energy has dissipated, and the cycle repeats.