Explanation of Operation of Class E Amplifier Circuit
Yep! The switch effectively controls the output waveform, both in terms of frequency and amplitude. The DC pulses it generates are smoothed into AC by L1 and C1 and the output filter (the RLC circuit). Keep in mind the switch in the diagram is just a representation, in practice that would be some sort of digital switch controlled by a PWM driver. As for actual values for the filter components, that will depend on what you want the amp to do. 

FWIW, I assumed that by basic you're looking for a fairly simple explanation. If you want a more general (or specific) idea of any aspect of how class E amps work ask away.