This is just a test video to see what it's like. Sorry if it isn't that good but hopefully through suggestions I can improve! Also, I didn't plan out what I was going to say so it's not very good.
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