The Clamps were first introduced in Amateur Surgeon 3: Tag Team Trauma. Since then they have been used as a way to close burst arteries.
Amateur Surgeon 3: Tag Team Trauma Edit
The instructions on how to use the clamp are first shown in the surgery for Ed Scape. To use a clamp, one must tap onto the artery(more specifically, the bright colored line) in order to place a clamp down, where it must be fastened by rotating the crank. The player can then cauterise the artery to heal it.
A minor drawback with the clamps was that if not handled properly, it would be removed from the artery entirely, harming both the patient and your score.
Amateur Surgeon 4: Re-Generations Edit
The clamps have the same purpose, although they seem to be much easier to use in this installment (most likely because severed arteries during limb transplants don't flail all over the place like the previous game.)