Based on the forum/answers search on "Asset Store Download" it looks like I'm not alone. I purchased a texture set from the store, and when I went to download it I was informed that I'd need the Unity Editor to do so.
I use Linux for my main machines, so I tried using the Unity Editor on a Mac I have available. It must be too old as the editor wouldn't open at all. Next I tried a VirtualBox Windows 7 machine, and while I can see the Unity Editor, the downloaded file has an extension of "unitypackage".
I won't go more into the details, but all throughout this lengthy process I kept thinking in the back of my mind "Why did they make this so difficult to download what I just purchased?"
I have a Windows machine at home where I have setup the Unity Editor on there too. I'll try to import the asset into a blank project and then figure out how to export the textures.
Unity. Your revenue will probably increase if you make it easier for your customers to actually receive the product they buy. Right now I've spent $20 on something I can't actually get to.