Why does the video stop when I lock my iPhone or leave the app?

Background play is part of SkipTube Pro. On the free plan, SkipTube pauses the video when you lock the phone or leave the app. It keeps the time, so you carry on from the same second.

With Pro, the video keeps playing with the screen off.

If you bought Pro and it still pauses, check that you are signed in to SkipTube. Pro stays with your SkipTube account. Signed out, the app is on the free plan, even on the phone that bought it. See Background play.