When Scout says that Aunt Alexandra slips right into Maycomb society, but does not fit into her world, she is talking about a definite generation gap in education, communication and understanding. First of all, Scout is a tomboy who likes to dress in overalls, beat up boys, and hang out with Jem and Dill during the summer. Aunt Alexandra never was a tomboy and was brought up wearing dresses, attending tea parties, and always doing what society expected of a young, privileged white girl. Based on how each of them were brought up, there are bound to be differences of opinion. What Scout doesn't know from her upbringing is how to behave like the privileged, Southern white girl that she "should" be. Aunt Alexandra, then, is asked to move in to help with the kids, and she focuses on teaching Scout everything she knows about being socially accepted in an age-old society. Scout clarifies Aunt Alexandra's way of life as follows:
"Aunt Alexandra was one of the last of her kind: she had river-boat, boarding-school manners; let any moral come along and she would uphold it; she was born in the objective case; she was an incurable gossip . . . She was never bored, and given the slightest chance she would exercise her royal prerogative: she would arrange, advise, caution, and warn" (129).
In these ways Aunt Alexandra fits into adult world of Maycomb's women like a glove. Scout and Jem, on the other hand, do not have any of these things in common with her. The generation gap is large, mostly because their mother died when they were very young and they were not exposed to such teachings for a long time. All they had was Calpurnia, who is black, and couldn't have invited white women over for tea. Atticus worked and had no time, interest, or understanding of the world of adult women in Maycomb, so he couldn't teach Scout the ways of women's society. Scout's amazement that Aunt Alexandra fit like a glove into Maycomb's social world creates a lesson by itself as she watches that world come into her home little by little. For example, Aunt Alexandra often hosts missionary tea parties. By doing this, Aunt Alexandra shows Scout by example how to be a hostess, make the refreshments, and behave in polite society. Eventually, Scout is given the chance to participate in these feminine gatherings, but it takes some time for her to learn about it.
In the end, it is Scout who must grow into the world of adults, but Aunt Alexandra will never be able to live in the world of a child. And since Aunt Alexandra has no concept of how Scout and Jem have been brought up so far, she cannot relate to anything they do or say. All she can do is contradict them and try to teach them her ways.
Aunt Alexandra comes to spend some time with Jem and Scout. The trial is going to start soon, and Atticus wants Aunt Alexandra's help with the kids. Jem and Scout are less than happy to learn that Aunt Alexandra is coming to stay with them. Aunt Alexandra is from the old school of the South, where people believed that a person's social standing was based upon how "good" their families were. This is in direct opposition to what Jem and Scout believe.
"Aunt Alexandra fitted into the world of Maycomb like a hand into a glove, but never into the world of Jem and me. I so often wondered how she could be Atticus's and Uncle Jack's sister that I revived half-remembered tales of changelings and mandrake roots that Jem had spun long ago."
Atticus has taught Jem and Scout to judge a person on the way they treat other people, and that a good family does not make a person good or bad. It is the person's character that makes them a good person. Aunt Alexandra is very caught up in the society thing. She believes that a family of good standing is what matters, and nothing else. Atticus, Jem and Scout know better.
The answer to this question can be found in Chapter 13. To understand why Scout says this, you should look at the paragraph above the lines that you are asking about.
The reason that Aunt Alexandra fits in to Maycomb is that she is very conscious of people's families. She thinks that a person's place in society is determined by their family. In the previous paragraph you can see a bunch of examples of how she judges people based on their families.
But this is not a good fit for Jem and Scout. Atticus has taught them not to judge people based on family and not to be arrogant because they are from a "good" family.