Have you registered that Kindle app with your Amazon account? If you have, then there should be an Archive button on the app's homescreen. Pressing that will take you to your account on Amazon, where you can download any/all books you have gotten from Amazon.
The app itself is not a book search app. It's a direct line/feed to Amazon. In addition, to actually buy books at Amazon, you have to go to Amazon via a web browser, you can't purchase them in-app any longer. This also means you can't browse within the app itself for books availabe for sale on Amazon. That function was removed late summer of this year...
Hope this helps.
Marilyn