To add Amazon Prime Video to your lineup of apps, you'll need to have a 3rd Generation Apple TV or newer, and have updated to the latest tvOS. Search for "Amazon Prime Video," and tap to download. The app also supports Universal Search and will be integrated into the Apple TV app, which makes both finding what you want to watch and discovering things you might like much more convenient.
Early reviews are somewhat mixed, with some users claiming the app's interface is a bit clunky and tough to navigate. Then again, a crappy movie menu page is still a whole lot better than having to sync three separate devices just to watch the latest season of Transparent.
h/t The Verge