Last month YouTube announced that they were going to introduce an offline viewing option to their mobile apps sometime in November. They've since rolled out details about how it's going to work. If you add videos to your device and then disconnect it from the Internet, you will be able to watch the content for up to 48 hours. After those two days have passed, you will have to reconnect in order to watch the videos again, but the 48 hour window will refresh, and the content will remain on your device.
Offline videos and playlists will be stored under an "on device" section of the YouTube sidebar, and you can add more content from the watch page of whatever you're viewing. Unfortunately we don't have screenshots to share, as the updated version of the YouTube app is still scheduled for a November debut. In the meantime, here's a video of a cat playing with a goat.