The working principle of this oscillator is as follows:
We have a classical tank circuit (L5||C1) which is connected in a closed loop with a two-stage non-inverting amplifier. This amplifier consists of an emitter follower (Q2) driving a common base amplifier stage (Q1). Hence, there is no phase inversion within the loop (condition for positive feedback).
The gain around the whole loop (loop gain LG) satisfies the oscillation condition (LG>1) at one single frequency only. This frequency was given already in Andy aka´s answer.
Remark (edit): The large value of the coupling capacitor C3 has no influence on the oscillation condition (at the frequency range of interest) and the value of R3 determines the feedback factor - and, thus, the magnitude of the loop gain which should be (slightly) larger than unity (0 dB).