You may be treated to a new message the next time you open the YouTube app. It prompts you to try the new Dark Theme with a button to enable it immediately. If it works for you, you'll see basically everything but the video thumbnails flipping from light to dark, or in the case of the buttons, dark to light. Overall, it's just much nicer on the eyes.
However, for many people, hitting that button to try the new Dark Theme is going to do nothing. In many cases, people have been able to get it working by force closing YouTube and restarting, but even that has been hit and miss.
If you're stuck in this position, first check in Settings -> General for a Dark theme toggle. If it's not there, I got it working by clearing the storage on the YouTube app, starting it and waiting a few minutes while it pulled data from the YouTube server, then did another force close and restarted the app. At that point, the "Dark theme" toggle finally appeared.
Now, go forth and use YouTube in the dark like humans were always meant to. This is a great day indeed.
Update: It looks like version 13.35 may be required for the Dark Theme to be available. Here's a link to sideload it if you can't get this update from the Play Store.
The APK is signed by Google and upgrades your existing app. The cryptographic signature guarantees that the file is safe to install and was not tampered with in any way. Rather than wait for Google to push this download to your devices, which can take days, download and install it just like any other APK.