Can someone explain to me how this works and what brand would be best
to buy?

http://bimmerforums.com/forum/showthread.php?t=158378

Clay is supposed to remove material that is embedded in the paint. Stuf that washing will not remove. It pulls the material out of the paint or shears it off at surface level.

Mother's or Claymagic are good kits. Lots of other good brands out there, too, I suppose. You want a fine clay. I understand that there are some clays that are more coarse, but you're not likely to run into those becuase they're mostly used by body shops, I think.