Sorry for the jumpy video in some parts my camera was having a bad day and I had tried to do this video 3 times and it still kept sticking so I thought I would upload one anyway. :)
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