To understand the answer to this question concerning "A Very Old Man with Enormous Wings" by Gabriel García Márquez, it is important to go back to the beginning of the story to understand why the old man appears in the first place. The family finds him at the end of three days of heavy rainfall. He has collapsed in the mud of their yard and is too weak to rise. It seems that the weather has weakened him and damaged his wings. Instead of caring for him, the family locks him in their chicken coop. They charge neighbors admission to view him, and the neighbors throw rocks at him, tear off the feathers on his wings, and torture him with a hot branding iron. These circumstances would have made the old man's condition worse instead of healing him.

The old man stays with the family for so long because he is elderly, sick, and crippled. By the time the chicken coop collapses, the old man can't go anywhere; he cannot walk, and his wings are all but devoid of feathers. He can only drag himself around the house and yard.

Note, though, that the old man with wings only stays as long as he has to. When he feels that he is regaining strength and his feathers are growing back, he goes off to a corner of the yard where the family cannot see him. Perhaps he is afraid that if the family notices his recovery, they will try to prevent his escape. So he bides his time until he is fully healed, and he flies away, presumably to his homeland.