There are a number of ways. I think the most obvious one is in the final chapter when Scout walks Boo home. She stops for a moment and examines the window through which he watched much of their lives. During this momentary reflection she begins to imagine the world through Boo\'s point of view. She sees the seasons change and the kids grow and relives all the their exploits that Boo was able to witness. By doing this she can understand how protective Boo would be of them since they offered him a way to experience the world, albeit vicariously.
Another example is after Scout \'rescues\' Atticus from the lynch mob that arrives at the local jail to get Tom Robinson. She shows the mob - in Scout\'s own innocent way - what Atticus has done for all of them. She makes them recognize their own humanity. So it is no surprise when we learn that one of the Cunninghams is on the jury and battles for an all out acquittal of the charges
Atticus too stands in the shoes of Bob Ewell. When he utters the great line \"I wish Bob Ewell wouldn\'t chew tobacco\" at the start of chapter 23, he calms Jem\'s fears by telling Jem to indeed put himself in Mr. Ewell\'s place. And Atticus calmly and rationally explains why Mr. Ewell hates him. Atticus doesn\'t judge him or prosecute him. He just understands Mr. Ewell\'s anger.
One scene that vividly illustrates this quote is at the very end of the novel where Scout walks Boo home. She stands on his porch after dropping him off, and envisions everything from his perspective. Throughout the novel, she has progressed from fear of Boo to fascination to a final mature understanding of where he might be coming from. Physically standing on his porch allows her to literally and figuratively be "in his shoes", and as she does she feels an understanding and a compassion for Boo.
Significantly, the novel also shows the problems in Atticus's advice, that he doesn't completely subscribe to this full empathy. When Scout asks him if it's OK to hate Hitler, he says no. However, the text suggests that Atticus is able to have intense dislike--I hesitate with hatred--for Bob Ewell, calling him the lowest sort of human being because he took advantage of someone like Tom, injured on the one hand and black on the other, therefore very low on the Maycomb hierarchy of value. Eliminating Ewell from the town is depicted as similar to eliminating the rabid dog--at the end, he is comfortable with having Ewell dead. Despite the evilness of that man, he is a human being, and I don't find Atticus finding opportunities to "walk in his shoes."
The author uses several events in the story to illustrate this. When Jem gets in trouble for bothering the old lady across the street, he is at first angry with his punishment. He cannot understand why his is not allowed to be nasty to her when she is nasty to everyone else. His punishment is to read to her every day and by doing this he gets to know her and see the daily pain she has to live with. Therefore he gains understanding.
Additionally, the children learn through experience not through school. By knowing their black neighbors the children learn that many of the things which the white adults in the neighborhood say about blacks is incorrect. By getting to know Tom and his family, the children are able to better understand the situation Tom was in.
Finally, Boo Radley. Initially, the children are afraid of him because he is a recluse. They do not know him. Boo leaves toys for them in the knot of the tree and eventually is he hero of the story when he saves the children from Bob Ewell. They no longer fear him but also have pity for him and understanding. By getting to know him too, the children can imagine his situation and 'step into his shoes'