How does this signal chopping circuit work?
You just have to treat the MOSFETs as switches. Whatever voltage exists between the power supply and the signal driver on the left is imposed across the primary of the transformer, first one way and then the other, just like any other H-bridge.

This creates an AM signal, in which the amplitude is controlled by the signal, and the square wave "carrier" is created by the MOSFET driver circuit.

On the right side, the switches (MOSFETs) are driven synchronously with the ones on the left (the exact synchronization mechanism is left as an exercise for the reader!), and this has the effect of full-wave rectifying the AM signal on the transformer secondary, which produces a replica of the waveform of the original analog driver signal. A little low-pass filtering will remove any remnants of the carrier frequency.

So, the FET driver circuit on each side must produce gate waveforms that switch the MOSFETs between fully off and fully on in the correct pattern, and the switching on both the "transmit" and "receive" sides must be synchronized.