YouTube is testing an official way to download videos on your desktop web browser (via Android Police). If you want to see if you’re eligible for the test, which runs through October 19th, check out YouTube’s experimental features page, which lists tests available for Premium subscribers. I was able to opt in to the test, but a colleague of mine wasn’t, which seems to indicate that not everyone is able to try the feature just yet.
If you’re opted-in and on a supported browser (“the latest versions of Chrome, Edge, or Opera,” according to Google), when you’re watching a video, you should see an option to download the video under the player.
When you click it, YouTube will download the video, which you can then watch from the Downloads section that’s accessible from the hamburger menu on the left side of the screen. I can’t find a way to download videos and watch them outside of my browser — this seems to be more of a way to watch YouTube videos offline.