YouTube recently updated their Android app, and one of the changes is something that will only really affect YouTube Red users. But apparently, Google has decided that it would be a good idea to remove the "x" that used to appear in the background playback notification that kills playback. You can pause playback still, but you can't get rid of the notification. What this means is that you need to go back into the app, exit the video, then exit the app to remove the notification from your notification shade. A bit of a step backwards in usability.
The reason why this only really affects YouTube Red subscribers is because, background playback doesn't exist for regular YouTube users. It's one of the few features that YouTube touts with YouTube Red - which costs you $9.99 per month and is bundled with Google Play Music Unlimited. Users had been asking Google to implement background playback - especially since so many podcasts are available on YouTube - and now Google has just made a bit more difficult to get rid of that notification after you're done listening to that video. This change will likely be reverted in the next update to the YouTube app, though, there's no hint as to how long that will be.
Right now, there is a workaround - sort of. If you don't use background playback at all, then it's probably a good idea to turn that off in the app settings. Probably a good idea if you don't have unlimited data, especially if you are watching YouTube on a mobile connection instead of WiFi. But if you do use background playback, then you'll have to settle for simply exiting the video before you exit the app. Otherwise, you'll be heading back into the app to exit the video properly. Currently, it's unclear if this is a server-side update, or a change that was made in the latest update to YouTube. Either way, Google can likely fix it without needing to push out a new APK to everyone that has YouTube installed - which is just about everyone with an Android device since YouTube is pre-installed on every device with Google services.