According to the app’s description, “Browse, Search, get product details, read reviews and see immersive product images and videos for millions of products available from Amazon.com and other merchants. Prime membership is required to make a purchase on your TV.” However should note that this is only a shopping app, so access to other Prime features are not available right now.
This means that if you were hoping that you’d be able to watch Prime Videos, you’d be out of luck. If you might recall earlier this year Amazon said that they would only bring back the Apple TV to their website under “acceptable business terms”. It was speculated that this is because Amazon did not want to sell a competitor of the Fire TV.
An alternative is that Amazon isn’t too happy about the 30% cut Apple takes from in-app purchases, which is why they have removed the ability to make in-app purchases from its apps and other platforms like comiXology. In any case the app can be downloaded here if you are interested in checking it out.