Here's my (very simplified and probably somewhat flawed) explanation of how it oscillates:
Ignore Q2 for the time being, and consider only C4, L1, C5, and R7. When power is first applied, C4 and C5 charge through R7 with their tops positive, and meanwhile the current through L1 increases as it builds up its magnetic field; but as C5 charges, the current through it decreases. L1 resists this and its decaying magnetic field continues pushing current up the bottom of C4 and it becomes positive with respect to its top. The current through L1 goes to 0 and then reverses as C4 begins to discharging through it. Thus it should be clear that there will be oscillations, but owing to natural resistance and the energy lost through the radiation of the antenna, they will eventually die out, and a steady state will be reached where there is no more voltage across C4 and L1, and C5 will have charged to the full supply voltage.
Now we can see what Q2 adds during each cycle of oscillation: when C4 and L1 are in the "clockwise" direction of the cycle, i.e. C4 is charging from the bottom, the top of C5 is positive and this reaches the emitter, keeping Q2 off or nearly so; but on the other half of the cycle, when the current through C4 and L1 is "anticlockwise", near the end of the cycle the top of C5 is pulled negative and this turns on Q2 by pulling its emitter down, causing more current to flow and thus replenish the energy of oscillation, causing it to continue indefinitely.