Let's read about some lovely trains! We won't be reading about the one in the thumbnail, unfortunately. That's one that my younger son built out of Legos several years ago. I apologize to train enthusiasts everywhere if I mispronounced anything. Don't take it personally. Be like a train and keep on chuggin' along. :)
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