It's out of sync a little, I know, but I wanted to upload something! I'll probably end up fixing it and re-uploading at some point, I just didn't want to disappoint you guys!
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