This is an interesting experiment, and I'd like to do my own tests with a VNA and/or GNU Radio to see the waveforms in detail. Regardless, here's what I think is going on.
If you look at a block diagram for an FM receiver and an SSB/CW receiver they are almost identical at the front end.
I think what is happening is that the CW carrier being introduced is making it past the mixer and filter(s) because it is acting a bit like a BFO. This is also suggested in the video when the injected carrier is detuned slightly to improve overall reception. This allows the two carriers to mix, and if you can detune one just right the phases when mixed will tend to add instead of cancelling, which lifts the signal out of the noise. And since the intelligence is an interrupted carrier, if you can hear the carrier, you can hear the intelligence.
That is, the two carriers are allowing a sort of AM style demodulation with a "recovered" carrier before the frequency discriminator stage, where the actual frequency variations are turned into intelligence. This all happens in the front-end, of course, where the radio is just recovering the RF into something it passes to the mixer.
Basically, I think we are taking advantage of the fact that a strong local signal can cause cross-modulation in an AM signal. This is basically an example of RFI being used on purpose.
The injected carrier, being FM, is also probably at a rather low power to reduce the capture effect, where a strong FM signal near or at the same frequency will cause only the strongest to be demodulated. Detuning the carrier frequency probably helps with that, as well.