The coil doesn't oscillate, the entire circuit oscillates.
If you use the proper ferrite core then there aren't any eddy currents to waste power.
The resistor makes the transistor conduct. The increasing transistor collector current in the coil causes the transformer action to also increase the transistor's base current.
The transistor conducts as much current as it can then the increase in current stops, and also the transformer action stops which causes the transistor's base current to reduce and then the transistor's collector current in the coil also reduces and transformer action futher reduces the base current until the transistor stops conductiong and the transformer action also stops. Then the resistor starts the transistor conducting again.