How Colprit Oscilator Satisfy Barkhausen criterion?
The Oscillation Criterion (Barkhausen) requires that the magnitude of the loop gain (turn-around gain within the complete feedback loop) is somewhat larger than "1" (0 dB) with a phase shift of exactly 0 deg (-360 deg) at one single frequency only. In the shown circuit this is accomplished by the amplifier (common emitter stage with 180deg phase shift) and the L-C combination in the collector path (additional 180 deg at resonance). Feedback to the base is accomplished by the capacitor C3.

However, each oscillator with a loop gain >1 will generate a sinusoidal signal with rising amplitudes with hard-limiting caused by the fixed supply voltage. As a consequence, the signal will have much distortions (bad THD). For this reason, it is wise to incorporate an amplitude-sensitive part or circuitry which causes "soft-limiting" of amplitudes. This will automatically reduce the loop gain  to the theoretical (ideal) value of "1". For this purpose, you can use diodes, thermistors, FET`s (as resistors) or other non-linear (amplitude-sensitive) devices.