In answering this question Could Wolverine's claws be prevented from unsheathing if his wrists are bent? I couldn't find anything that actually described what caused Wolverines claws to pop out. I've seen pictures of x-rays and there doesn't seem to be extra bones to help. I would assume that there are extra muscles involved but this is never mentioned. His claws are a part of his mutation not surgically inserted like it was previously believed. So what makes them work?
I'm not sure if this is canon, but I have found this drawing here. Is show a nice cut out technical drawing in the center that show some muscle attached to the claws to move them.
This might be better as a comment on someone else's answer, but not sure which one. This exact topic was illustrated in an infographic by Daniel Mora, who is a Costa Rican artist rather well-known for his Marvel & DC art.
You can view the entire infographic here, but the relevant part is snipped out & shown below. The original is in Spanish, but I've done a rough translation of the text on this part:
Was going to put this as a comment in reply to Kevin Howell, but it will be longer than a comment will probably allow.
Speaking from a physiology standpoint, it is going to probably act much the same as the muscles do for the leg. Your quadriceps on the front bring your leg from a bent angle to a straight angle, and the hamstrings in the back do the opposite.
So I would surmise that (surgically rearranged or not), there are opposing sets of muscle/tendon complexes (tendons connect muscle to bone, you have anchoring tendons at each end of a muscle). One set relaxes as the other contracts, and the claws slide out. To retract, the opposing muscle contracts as the first one relaxes.
This would also give a good explanation of how Wolverine is able to control the speed and individuality of the claws, much like you can make a slow or fast fist, or extend different fingers apart from each other. (With some exception in various positions for the middle and ring fingers as they share a tendon).