In To Kill a Mockingbird, Atticus says that the best way to understand a person is "to stand in his shoes." How is that illustrated?

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'