Understanding Working of LC Meter
Basically the "classic" relaxation oscillator only works as a kick off to start the LC oscillator.   

The little damped ringing on non inverting input triggers the output to swing full supplies and increases itself.
 In the frequency domain you may think that parallel resonating LC tank gives zero phase and hence positive feedback at its resonance frequency only.  

Now the "slow" RC network on inverting input simply sets the threshold at the average output value to ensure proper operations even with non symmetric (both in level and time) output