An example would be a streamer that can watch his/her chat while playing a game without needing a second monitor or phone/tablet.
An other use for a feature like this could be a user being able to put a picture of something like a map or capture part of a tutorial/guide to a pop-out window while playing a game.

I can also see this feature possibly increasing the user's productivity levels with the ability to capture a part of a window, move it around with ease, having it float over whatever you're doing and still being able to click the window underneath. No overlapping of different tabs and using alt+tab to bring back the window you pull your information from.

The way I would use this feature the most is - As a moderator of a relatively popular Twitch.tv streamer, having the ability to watch the streamer's chat while in a game, would be extremely useful. I wouldn't have to purposefully move my eyes all the way to my second monitor and finding out I've missed a message that should have been removed from the chat with a delay.

I'm not sure if exactly this would be the use case scenario for most people, but it can give you an idea of what I'm trying to suggest.