The other answers are good, but in case you are asking a more basic question about how GET is being used. . . GET is a very interesting word, with many different meanings. In this case it means something like 'To cause to happen / convince". So it means to convince/cause/motivate/ etc. an animal to eat out of your hand. Similar examples: I wanted to sleep late, but my mother got me to wake up early and study. He's lazy, but I'll try to get him to clean the house.

A side note. That's "Because of Winn-Dixie," by Kate DeCamillo. It's a terrific book. Warning: it's easy reading for a ten-year-old US native speaker. It's not so easy for a foreign speaker because a lot of the dialog is informal. It is full of idioms, phrasal verbs, and so on. Some of it is even regional and specific to part of the United States, and to US culture. Keep posting your questions!

Otis is scared. Otis is shy. Otis is afraid of social situations like parties. Imagine you want to tame a wild horse. The horse will be shy. It will be afraid of people. What do you do? You offer it food it loves, like an apple. You want the horse to come close to you, so you try to get it to eat out of your hand. You talk gently and soothingly to it. She is trying to make him feel secure, the same way you would make a wild horse feel secure. She isn't going to feed him out of her hand, of course!