To "ring true" literally refers to test a silver coin by dropping it onto a flat surface and listening. A real silver coin--sorry, I can't vouch for this myself!--is supposed to have a clear, bell-like sound, while various fake, impure, or plated coins sound different. Thus, "that doesn't ring true" means, literally, it doesn't have the right sound to the ears. So figuratively it can mean "that doesn't sound true" or "that doesn't seem true."