It's a great feature for music videos, podcasts, ambient sounds and it's unfortunate that Android doesn't include it and Apple (accidentally?) disabled it in iOS 6.1. Until Apple fixes the bug, you can install two third-party YouTube apps that use some undocumented APIs to support background audio: McTube and YouPlayer. McTube continues to play the video after closing the app, while YouPlayer lets you manually enable background audio.
YouTube Rewind: What you watched in 2013
5 minutes ago