I don't really use Snapchat, and when I do it's just to take private pictures that I never upload. These I took in the past. My Snapchat is messed up now anyways so I can't use the filters anymore.
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