YouTube has begun a wider rollout of offline video downloads for its Android app in countries that didn't previously have it. While it seems to be popping up in parts of the Middle East in countries like Lebanon, India, and Iraq, it also seems to be showing up here in the U.S. for some users, though it's also been reported that some U.S. users don't have it, so it still seems to be going live for some at a slower pace.
There doesn't seem to be any rhyme or reason for Google's chosen location rollouts so if you're curious to see if you have the option to download YouTube videos for offline viewing later on, simply open up the YouTube app on your mobile device, go to any video as it won't matter which one, and in the top bar of action buttons just under the title you should see "download" as an option just to the left of the "add to" button. When you tap this button for the first time you'll be asked to select your download quality preferences for videos, and there will also be an option to have the app remember your settings for future downloads.
If you check this box and you decide you may want to change the option later, it's possible to do so from the settings menu. From the app settings, scroll to find Background & Downloads and in here there will be the option to set the download quality as well as the toggle to download over Wi-Fi only and a few other settings. You can also access your downloaded videos from the Library tab in the bottom navigation bar. Downloads will be kept on your device indefinitely, unless you go a period of 30 days without an internet connection after which the videos will be automatically removed from your downloads list. The kicker is that as long as you re-connect to the internet at any point within that 30 days, even if it's already been 27, that time will reset and you'll get to keep the downloads still. This may or may not be showing up for other users in the U.S. currently but there should be a popup message coming from the download button on the video playback page if the option is live for you.