This question is not fully explained in To Kill a Mockingbird, but it appears that Alexandra felt that her feminine touch was deserately needed in the Finch household. It is clear to Scout, however, that Atticus has not requested her presence:
I had an idea... that Aunt Alexandra's appearance on the scene was not so much Atticus's doing as hers. Aunty had a way of declaring What Is Best For The Family, and I suppose her coming to live with us was in that category.
Atticus probably found it impossible to discourage Alexandra from moving in, and he must have thought that she would be useful for a few weeks, since he was going to be busier than usual dealing with the Tom Robinson trial. But Alexandra was dismayed at Scout's unladylike ways, and she thought her brother was doing a poor job of raising them. Alexandra thought she would best serve as the mothering symbol in the household, although Atticus was perfectly happy with Calpurnia serving in the same mold. It is clear that Alexandra was far from the perfect mother: Her own son had left home as soon as possible; her grandson, Francis, was a spoiled brat; and she and her husband, Jimmy, barely spoke to one another. But Alexandra believed Jem and Scout needed another Finch female in the home--she considered Calpurnia only an employee and not a member of the family, like Atticus. She believed the children needed lessons in "gentle breeding" and the Finch family heritage, so she left Finch Landing for an extended stay--and never left.
Why does Aunt Alexandra come to visit in To Kill a Mockingbird?
According to the previous poster, her son "left home as soon as possible," her grandson "was a spoiled brat," and husband "barely spoke" to her.
She must have come to visit Atticus to get away from her own life.