Wireless power transmission circuit component functions description
Here's a very similar circuit: -



Stolen from here and this is the explanation (which I agree with): -


  The circuit works by one mosfet turning on due to differences in the
  gate resistors or internal structure of the mosfet. Once on, the
  opposite mosfet will be held off by the fast diodes. The voltage
  across one primary half will rise up an fall again in a half-sine
  wave. Once at zero the mosfet that was on will be forced off, and the
  mosfet which was held off will be allowed on. The cycle repeats in
  opposite this time, before returning to where it started. The large
  inductor serves as a "current capacitor", providing constant current
  to the driver. Thanks to the resonant action of the circuit, it
  benefits from ZVS, or Zero Voltage Switching. This means that the
  mosfets switch on with no voltage across them, so while they
  transistion from off to on they won't dissiapte power. (P = I * V)


The above uses one RFC at the centre tap of the transformer. The circuit in the question doesn't use a centre tapped transformer hence needs two RFCs. The RFCs are acting as current sources to develop the AC signal upon - they bias the transistors without acting as too much of a load for the switching AC waveforms on the actual transformer.

Here is a link to another stack exchange answer I gave on ZVS drivers that may be of interest.