Like the previous answer states, there is no one place where it says that Scout grew up. However, it is implied. If you look at the facts of the novel, then it is easy to say that Scout matured.
First, Scout experienced many difficult situations, which required maturity or in the least caused her to mature. For example, she sat through the trial of Tom Robinson. As such she faced issues of rape, family abuse, and injustice.
Second, she also experienced the brokenness of the world directly, as Bob Ewell tried to kill her and Jem. We can say that the innocence of her childhood at that point was shattered. She now knew that people could stoop very low.
Third, arguably the best evidence for Scout's maturity is when she was able to see that Boo Radley was a mockingbird, that is, a person who did not harm anyone. In fact, she was able to see this truth better than her father, Atticus. From this perspective, she was more mature than Atticus.
While there is no succinct statement, "I have grown up," by Scout, the reader can infer in two parts of the novel that Scout has "grown up after the trial" :
(1) In the exposition of the novel, it is an adult Scout who narrates the tales of events after the Tom Robinson trial:
When enough years had gone by to enable us to look back on them, we sometimes discussed the events leading to his [Jem's] accident.
and (2) in the final chapter as Scout stands on Boo Radley's porch after having walked him home, she notes how her neighborhood, always so familiar to her, yet looks different from the perspective of the Radley's porch.
Atticus was right.....you never really know a man until you stand in his shoes and walk around in them. Just standing on the Radley porch was enough.
Then, as Scout walks home in the misty night, she reflects,
The street lights were fuzzy from the fine rain....As I made my way home, I felt very old....
Scout feels older because she has matured; she has learned to perceive things from another's perspective, and has, indeed, grown up.