Does the AppleTV had a place where I can see what's been downloaded to the device?
Nope, the latest versions of the AppleTV have 8GB of storage however the device doesn't allow the user to view or modify anything about the status storage (at least in a stock state).

During the introduction of the first all black AppleTV Steve Jobs said,


  People don't want to think about managing storage, they just want to watch movies and TV shows.