The Protagonists wife died.
In their culture they hold sky burials (which as
@madthejoint pointed out are a real thing).
He was supposed to proceed with the tradition of the sky burial, by having his deceased wife's body being eaten by the vultures. The old man warned him, that if he takes
any part of his dead wife with him she won't be able to go to heaven and turn into a vulture herself.
While cutting her up (in preparation for the sky burial) he saw that she was pregnant. Unable to leave the fetus behind he took it with him. Which
is a part of his dead wife's body.
So she became a vulture and came to take him.
But it is heavily implied that the protagonist was insane. He was seen as impure (any child he touches will be cursed). The old man (his father) said, that they performed a sham sky burial, so not a proper burial and the biggest clue: his sister-in-law didn't know if he came from a funeral. Which would be quite weird if his wife had just died.
The name of the story is also 'The pregnancy of the Kalaviṅka'.
Kalaviṅka is also the name of a buddhist immortal creature, with a human head and a bird's torso. It preaches the Dharma (in buddhism the cosmic law and order/ultimate truth)