Shake to start!
That crazy number of MOSFETs is nothing special. It's just a push-pull output stage. There are 3 NMOS in parallel and 3 PMOS in parallel.
This circuit is not really how a commercial inverter works. This is really just a Wien bridge oscillator, whose output is stepped up. There is no feedback from the secondary side back to the primary. There is no output or input protection. The output voltage starts to drop if the current draw is increased. THIS CIRCUIT CANNOT BE BUILT AND USED IN THE REAL WORLD.