Mrs. Peter undergoes a very interesting transformation throughout the story. First, she is not a personal friend of Minnie's and has no clue about the suffering that this woman had undergone by her husband. As the wife of the sheriff she holds a responsibility (just like the whole town does) to serve and follow the directions of her husband. For this reason, when she entered the house at first she was weary, doubtful, and almost hesitant to even consider Minnie's side.
However, as she saw the little details, those "trifles" as the story suggests, she was able to connect the dots and, as a woman, she was able to connect with Minnie. She, herself had moments where she found herself with no support from her husband, and she had seen first hand what cruelty from men to women look like when she witnessed a boy ax down her kitten. Therefore, her change was that she went from one side to another: From witness to supporter and from a stranger to a person who completely identifies with the situation.