Google has been testing Incognito Mode on YouTube app for quite a some time ago. At last, the privacy focused feature is rolling out to the end users within the latest YouTube 13.25.56 app. In short, the new Incognito Mode replaces the currently existing Sign Out option from your account settings. 
We came to know that the server side roll-out of the feature has reached to many users using latest YouTube app on Android. In fact, the Incognito Mode is present on all of ours devices as well thankfully. Hence it seems, Google has done with the beta testing of the feature and is fully-baked for a wider roll-out.
Needless to say, once the Incognito Mode is enabled your app won’t collect and store your watch and search history. Well we most of us are familiar with it on Chrome and other popular browsers as well as on GBoard. The Incognito Mode on YouTube does the exactly same function and nothing much new.
The Incognito Mode can be seen on the Account section of the app. The long existed Sign Out option is bidding adieu from the YouTube as it is replaced with “Turn on Incognito” option situated at the bottom. Once you enable it, then you’ll be signed out from your Google account and a notification ‘you’re incognito‘ will be shown everywhere on YouTube screen. Note that, only the Home and Trending feeds will be accessible if on Incognito Mode.
If you had enough with the Incognito and want to disable it. Hit the Incognito avatar on top of the YouTube to see the Turn off Incognito option. When you turn off Incognito or become inactive, your activity from this session will be cleared and you’ll return to the account last used.
The latest YouTube 13.25.56, by now, should be available on Google Play Store. If not updated yet, click the Google Play link of YouTube app below.
YouTube : Google Play Store
