If you've ever used YouTube on the web, you've undoubtedly seen the "Up Next" autoplay feature in action. When the video you're watching ends, YouTube will give you a timer before starting up the next video from the suggested videos list. Interestingly, those seeing the feature pop up on Android are seeing it as a toggle button, which would make it easy to turn off the feature if you don't like it. Presumably, this is because users on mobile need to be more careful about data usage, but it's a feature that is not offered on the web, so if you don't like the feature there, it's more troublesome to turn it off.
As of right now, it appears as though the feature is being tested with a relatively small section of users and there's no way to force the feature to turn on. But, if that changes, we'll be sure to let you know.