The girl got excited her new toy

You can think of by, in this case, in the way she got excited, she got excited due to earn this toy.

Using "with", passes the idea of the feeling that she had when she acquires it, the feeling when she in his company, maybe playing with it, using it.

So, by is used to express the way that she got excited. And with, expresses the idea of his possession and the feelings due to it.