Understanding Royer oscillator (for wireless power transfer), (specially mosfet gate driver circuit)
The Schottky diodes, together with the capacitor and resistor form a 'snubber' suppressor. This is really just a low pass filter and is quite common when using transistors with inductive circuits.

To calculate the value of components in a filter, you must first know what frequencies you're trying to suppress. 
In this case, it would be to stop very high frequency RF ringing in the inductors from reaching the gate of the transistor. The values are seldom critical in this application and the values shown should work for most variations of the circuit.

Start with the resistor and work out what value you need for the gate current.
Then use the RC constant to calculate the value of the capacitor needed for the desired cut-off frequency.