It is a simple extension that understands the download queue file that Amazon delivers after a purchase is made. Linux Desktop integration is provided so that your web and file browsers associate Banshee with the download queue file.
Further improvements are on the way, but I've downloaded many albums in Banshee at this point (and subsequently burned a decent amount of money!) Of course, there are a number of free downloads that Amazon offers, which were good for testing. Unfortunately, I didn't think to look for these until I had finished the feature.
In addition to the integrated Banshee extension, there is a command line client called bamz. It requires Banshee to be installed for support libraries, but does not integrate with your Banshee library in case you are into that sort of thing. By default, bamz downloads the tracks into your current working directory with the following structure:
$PWD/<Artist Name>/<Album Title>/NN. Track Title.mp3.
I will be merging the Amazon MP3 support into the 1.6 stable series, and it is currently available in Banshee's git master branch. Enjoy!