Apple TV Desktop
If you need access to more than just the messages you've received, you can go to the Desktop section of the Apple TV app.
This works like a simplified version of our web app's Desktop view, letting you dive into all of the folders and assets you would find there. Selecting a production from the poster menu here opens up the top level of folder structure you have access to inside the production.
From here you can slide in any direction and click into a folder where you'll see two things:
At the top are any subfolders that you have access to inside the current folder. Beneath that will be any assets which have been uploaded to the folder itself. These assets are organized into their upload batches and if you go into a folder with more assets—
You'll see subfolders again at the top, but beneath those are assets laid out in rows according to their upload batches. From here, scrolling down and up brings you into the different batches.
And scrolling to the left and right on a batch lets you look through all of the assets inside it.
Pressing the menu or back button at any time will bring you back to the previous level of navigation; a parent folder if you're in a subfolder, or the top level of folders if you're in a parent folder, etc.
Playback in this works exactly the same as in the Home view, so head over there if you need a refresher!