How do I put a new wallpaper on my Android 4.3 device?
Dropbox is making it seem more complicated than it really is. Unlike on the PC version of Dropbox, the files you see in the Dropbox app aren't actually stored on your tablet. It's just a list of files you could download. When you view the file from Dropbox, it downloads it temporarily into Dropbox's private storage area, and then temporarily gives the app you choose access to the file.

Just click "Save to SD card". That actually downloads the file to your tablet, and  saves it permanently in a storage area that other apps have access to. After that, the image will show up in Gallery, probably in its own "Dropbox" album.