Can someone explain how this oscillator circuit works?
From startup the capacitor will be discharged so current will flow through the resistor, capacitor and LED, charging the capacitor.  When the cap voltage reaches the Vbe drop of the NPN it will start to switch on, also switching on the PNP and so current flows through the PNP and the LED.  In this state the capacitor is discharged via the resistor and the PNP.
There might be a stable state where the capacitor is just at the Vbe of the NPN, but the combined gain of the two transistors will make this zone extremely narrow and the circuit’s propagation delay will mean that it always overshoots from one unstable state to the other.