Justin is using a Universal OX system (he points this out at around the 6 minute mark) which converts the speaker output from the amp into a signal that can be fed directly into an audio interface.
After that, there’s a few options for monitoring with headphones: he could plug the headphones into the OX, or into his audio interface and do “direct monitoring” with the audio interface, or use software monitoring in his DAW (I believe he is using logic).
And the audio for the video can then be recorded in the DAW.
Cheers,
Keith