(IANAD) When they do surgery that requires stopping the heart, they divert the bloodflow to a machine (essentially an external mechanical heart) to keep the blood flowing where it needs to be flowing. That frees them up to remove the old heart (which is a complicated set of muscles and valves) and hook up the transplant heart. They sew up the connections, [and the heart has a group of cells] to tell the heart to pump. When the surgeons are ready, they re-divert the bloodflow to the new heart, which knows to beat because of its special cells.
Since the body recognizes that this new muscle is foreign material, the transplant recipient must take drugs the rest of their life to keep their body from attacking the foreign material.
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