It's a crystal receiver with a low frequency amplifier. The horn antenna is assumed to be good enough bandpass filter (=to keep out lower frequency signals) and of course it can catch enough power from a near or strong enough transmitter to generate a DC voltage by rectifying the microwaves with a diode. The diode is a Schottky diode designed to rectify at several GHz, no common Si-PN diode would work.
The amp is an ordinary opamp connected to a non-inverting amplifier. The gain is adjustable.
If the diode rectifies a pulsed microwave signal the amp output DC pulses, which can be heard as a sound, if one feeds them into a speaker. The speaker should be active one, an opamp cannot feed a speaker properly. Audible sound needs that the microwave signal is pulsed in an audible pulse repetition frequency.
A continuous microwave signal generates non-zero DC output. There seems to be an offset zeroing potentiometer intended for calibration the output to 0V when there's no DC input.
This receiver is not especially sensitive, but it can detect signals in many experiments. The linked system very likely allows experiments where moving target causes doppler shifts.
When Doppler shifted and the original signal from a continuous sinewave transmitter meet in this receiver at the same time, one can see how the DC output voltage varies just in the Doppler shift frequency. That's effectively the "frequency difference mixing result". A speaker may produce a sound of the Doppler shift. In the past there were land surveillance radars based on this effect. A well trained radar operator could hear what kind of target moved in the radar beam.
Communication radio links must have much more advanced receiver designs.