Royer Oscillator Circuit
There are several variations of Royer oscillator. The original version gives out squarewave. Its major idea is to have a saturating transformer to determine, when state flip occurs.

A popular variation is the one that has a capacitor in parallel with the transformer primary. This uses non-saturating transformer that acts as a LC resonator coil, too. The output is sinewave, Frequency is the resonance frequency of the LC circuit. 

When started, both transistors start to take current. That is  true for all variatons of RO. Componets are different. Thus the current draining race isn't well matched. The winning half cuts the bias off in the losing half. Thus only one transistor stays ON. The seesaw flips when the current reaches the saturation point. At that point the off-state transistor starts to get more bias and this is accelerated due the diminishing bias in the on-state transistor.

The coil inductances, operating voltage and transformer core saturation limit determines the  state flipping frequency.

For rf a sinewave output is wanted. By adding the capacitor the resulted LC resonance ringing unavoidably starts to  turn the current drawing tide as soon as the sine curve has reached the culmination point where the bias of on-state transistor starts to decrease. Altough transistors are switched on-off the output voltage resembles sinewave due the principle of LC resonance and its bandpass nature.

Transistors or mosfets in high voltage or high power ROs need some protection. Also their swiching speeds need some boosting. These demands are the most common reasons for additional circuitry. 

This all and much extra detail can be found in English Wikipedia under Royer Oscillator.