I know it's not obvious, but you actually can -- you probably just don't notice.
Some scenes are saved on your iPhone/Touch, including the 50 or so that came bundled with the app, as well as any scenes that you've subsequently selected. Scenes that aren't saved locally still show up in the scene browser, but with their thumbnails grayed out. Tapping on those downloads them, and you'll notice that the thumbnails turn solid once you do that; they should also give you a network error if you try to load them when you don't have an Internet connection.
Also, the app checks daily for new scenes on the server, and shows them in the scene browser.
Anyway, glad you're enjoying it!