Why did Aunt Alexandra come to stay with Atticus, Jem, and Scout and what does her arrival tell the reader and the people of Maycomb about her relationship with Atticus?

At the end of chapter 12, Jem and Scout return home to find Aunt Alexandra waiting for them on their front porch. At the start of chapter 13, Aunt Alexandra explains to Scout that because both she and Jem are growing up, she and Atticus believe that Aunt Alexandra's feminine influence will be helpful to Scout. Scout does not agree with this assessment of need, and when Atticus comes home and explains the decision, Scout gets the feeling that Aunt Alexandra had made the decision to come live with them all on her own.
Aunt Alexandra's desire to come and live with Atticus and the children may stem from a desire to be helpful during a difficult time and a genuine sense of familial duty to her brother, or it may come from a less generous impulse that serves Aunt Alexandra's needs more than anyone else's. Scout tells the reader that Aunt Alexandra ignores her question about missing Uncle Jimmy, so perhaps Aunt Alexandra is bored by her life at home and wants to get involved in something more interesting than her silent husband. Her arrival tells the reader and the residents of Maycomb that she is certainly interested in contributing to the upbringing of Jem and Scout, but her motivations to do so are less clear.

In chapter 13, Scout and Jem return home from their visit to First Purchase African M.E. Church and are surprised to find Aunt Alexandra waiting on their front porch. After Calpurnia takes Aunt Alexandra's bags inside, Scout asks her aunt if she has come for a brief visit. Alexandra proceeds to explain that she and Atticus decided that it would be a good idea for her to spend some time with the family in order to provide some "feminine influence" for Scout, who is rapidly maturing and will be interested in boys and clothing soon. Since Atticus's wife passed away, Calpurnia and Miss Maudie are the main female role models in Scout's life. However, Aunt Alexandra feels that she will also be a better influence on Scout, who enjoys wearing overalls and behaving like a tomboy. Aunt Alexandra's willingness to stay with her brother's family demonstrates her compassion and loyalty. Aunt Alexandra wants what is best for Atticus and his children, which is why she is willing to leave her home and stay with her brother's family to positively influence Scout.