I think the point of the question was the unusal use of "wax." My guess would be that it comes from the waxing phase of the moon -- if you "wax eloquent," your eloquence increases bit by bit. This is purely a guess, though.
I already looked it up in my dictionary. And I understand what it means, though I haven't used that word in conversation. But why with "wax"? I mean, what does "wax" have to do with being eloquent?