How does Coraline make it back home to her own world?

When Coraline realizes she is trapped in the world of the other mother, she understands she will have to use her wits to return home. First, she proposes a game to the other mother: if she, Coraline, can find the souls of the children trapped behind the mirror and find her parents, the other mother will free her, the souls of the children, and her parents. If she fails, she will stay with the other mother forever. The other mother agrees to this game.
Coraline is able to use her rock to locate the three marbles that hold the souls of the three trapped children. She realizes her parents are held in a snow globe, but she doesn't want the other mother to know she knows.
To confuse the other mother—and to get her to open the door between the two worlds—Coraline says she thinks her parents are in the passage between the two worlds. When the other mother unlocks the door to show her she is wrong, Coraline seizes the snow globe and throws the cat she is holding at the other mother. The cat attacks the other mother, and after a fierce struggle, Coraline gets through the door to the other side and back to her real home and parents.
In short, Coraline uses her brains, trickery, and surprise to confuse the other mother so that she can escape and save her parents and the souls of the other children.