I don't understand the working principle of this metal detector circuit?
It's based on the fairly common 555 feedback astable circuit.
this uses the output pin instead of the discharge pin in the timing circuit.

https://www.electronics-tutorials.ws/waveforms/555-circuits-part-1.html
this circuit has been modified with a capacitor and the sense coil inductor in parallel with the timing resistor  this allows conductors and magnetic materials that come close to the sense coil to change its inductance and that change the allow less or more current to flow changing the frequency