The circuit is a typical single (common collector) transistor colpitts oscillator. When current passed through primary winding, it induces a magnetic field to the core which in turns gives the energy back to the feedback winding with a time delay determined by the core material and winding. The feature of the colpitts oscillator is that the feedback for the active device is taken from a voltage divider made of two capacitors in series across the coil/inductor. The feedback coil, L and the series combination of C1 and C2 form the parallel resonant tank circuit which determines the frequency of the oscillator. Voltage across C2 provides feedback.
The frequency of oscillation is approximately the resonant of the LC circuit, which is the series combination of the two capacitors in parallel with the inductor or feedback coil. Approximately, oscillating frequency of the colpitts oscillator is shown below, it is however affected by strain capacitance in the circuit.