Where does the name come from?
See the excellent description at Wikipedia. It began with Sakura Systems which designed a high-end (or at least expensive) power amp based on LM3875 and a minimal power supply. The product is called Gaincard. Pictures are floating around on the internet and the inside is amazing I think. Some (rather many actually) are a little bit bugged (Rod Elliott among others and I for instance) over Gaincard but others, mostly DIY'ers build their own Gaincard and then call them Gainclone. Some of the Gainclones are really beautiful, very good craftmanship.
Here are reviews of the original.
http://www.tnt-audio.com/ampli/47gaincard_e.html
http://www.6moons.com/audioreviews/47labs5/gaincard.html
The beautiful things about the LM3875 and LM3886 are that they are reliable, cheap, sounds good and are easy to get working, even if the are hardwired (no pcb).