why does the albatross finally fall from the Mariners neck

The Albatross falls from him neck when he regains his ability to pray.

In Part IV, the Wedding Guest proclaims that he fears the Ancient Mariner because he is unnaturally skinny, so tanned and wrinkled that he resembles the sand, and possesses a "glittering eye." The Ancient Mariner assures him that he has not returned from the dead; he is the only sailor who did not die on his ship, but rather drifted in lonely, scorching agony. His only living company was the plethora of "slimy" creatures in the ocean. He tried to pray, but could produce only a muffled curse. For seven days and nights the Ancient Mariner remained alone on the ship. The dead sailors, who miraculously did not rot, continued to curse him with their open eyes. Only the sight of beautiful water snakes frolicking beside the boat lifted the Ancient Mariner's spirits. They cheered him so much that he blessed them "unawares"; finally, he was able to pray. At that very moment, the Albatross fell off his neck and sank heavily into the ocean.