It just means absolutely and completely sober, or not drunk. Someone who is stone cold sober hasn't had any alcohol, and is completely in charge of his body, his speech or what he says, and his mind. When someone touches a dead body, the body is as cold as a stone. It is absolutely dead and has no life in it. So stone cold is absolute, or all the way.

It means you haven't drunk any alcohol whatsoever. Sober can also refer to not having taken drugs. Stone-cold sober is an adjective, and it gives more emphasis than just "sober". Person 1: I can't believe you came to my wedding drunk! Person 2: Are you joking? I'm stone-cold sober. I have to drive home later.