The first example comes when Atticus explains this concept to Scout following her disastrous first day of school with Miss Caroline (Chapter 3).
"You never really understand a person until you consider things from his point of view--until you climb into his skin and walk around in it."
At the missionary circle tea, Atticus brings the news of Tom's death. Miss Maudie and Aunt Alexandra compose themselves, and then return to serving refreshments. Scout is impressed, and she suddenly recognizes that this is part of being a lady (Chapter 24).
After all, if Aunty could be a lady at a time like this, so could I.
Perhaps the best example is experienced by Scout herself as she stands on Boo Radley's porch, looking upon her neighborhood as if seeing it through Boo's own eyes (Chapter 31).
The final example comes at the end of the novel (Chapter 31) when Atticus carries Scout to bed on the night of Bob Ewell's attack. Atticus has been reading to her from The Grey Ghost, and Scout recognizes that Stoner's Boy, who had been accused of mischief in the story, is actually "real nice." The story parallels that of her own evening with Boo Radley.