We're not told specifically how Jim's story of 'Lizabeth changes Huck's view of Jim. The story Jim tells is about his daughter 'Lizabeth recovering after a bout of scarlet fever. Jim gets angry at her and knocks her down when, after she is well, she won't respond to his commands. Then he realizes she is not complying because the scarlet fever has left her deaf. Jim feels an outpouring of remorse for having struck her and expresses his deep compassion for her.
We do know, however, that Huck is beginning to see Jim more as a human being and less as a slave, because he draws Jim out to talk about his family. Huck overhears Jim lamenting the loss of his family:
He was often moaning and mourning that way nights, when he judged I was asleep, and saying, “Po’ little ’Lizabeth! po’ little Johnny! it’s mighty hard; I spec’ I ain’t ever gwyne to see you no mo’, no mo’!”
Huck believes that Jim's grief and longing show he is a good man. He thinks:
I do believe he cared just as much for his people as white folks does for their'n. It don't seem natural, but I reckon it's so.
This statement shows that Huck is beginning to think and reason for himself and break away from the social conditioning that has taught him all his life that slaves are not quite human.