This videos is SICK! Sorry, I couldn't help it. So it seems you've come down with a bit of a cold. Surprise, I've caught it, too! (This is an example of a person using their real-life misfortune in a useful manner. I actually am sick at the moment.)
To listen while using other apps, switch the player to Picture-in-Picture (PiP) during playback — it keeps playing in a small floating window (the screen stays on).
To listen with the screen fully off, in-browser playback stops by YouTube's design. Open the video in the YouTube app to keep listening where background playback is supported (e.g. with YouTube Premium).
Open in YouTube app