NOTE: I apologize for the shaky camera. I have no idea what was going on there. Sorry about that. Let's take a look at 73 of the rings from my ring collection!
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