Is your Apple TV up to date?
When I opened that app, I had three options at the top: Explore, Your Flickr, Search. When I selected "Your Flickr", I got the instructions to sign in to my Yahoo account at Flickr/Apple TV (a website address that was shown on my TV), and a code that I had to enter afterwards.
That's what I did in Safari on my iPad, and now I'm able to see my account on my TV.