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"First of all," he said, "if you can learn a simple trick, Scout, you'll get along a lot better with all kinds of folks. 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." (3.85-87)
Atticus's advice "to climb into someone's skin and walk around in it" is a little more Silence of the Lambs than the typical advice to walk a mile in someone's shoes, but the idea is the same: compassion is based on sympathy, on being able to put yourself in the other person's place and understand why they act the way they do even if you don't agree with it.
"This time we aren't fighting the Yankees, we're fighting our friends. But remember this, no matter how bitter things get, they're still our friends and this is still our home." (9.27)
If you're engaged in bitter warfare with someone, can you still be their friend at the same time? (Duh. That's why the word "frenemy" was invented.) But seriously—Atticus would say, sure can. You continue to treat them with a friend's respect, and you remember that they're part of a larger community that stays whole even if its parts are pulling in different directions.
"Scout," said Atticus, "when summer comes you'll have to keep your head about far worse things... it's not fair for you and Jem, I know that, but sometimes we have to make the best of things, and the way we conduct ourselves when the chips are down—well, all I can say is, when you and Jem are grown, maybe you'll look back on this with some compassion and some feeling that I didn't let you down." (11.53)
Sometimes it's kids rather than parents who just don't understand. Atticus knows that his behavior seems incomprehensible or just plain stupid from some perspectives, so he hopes Scout and Jem will be able to understand why he did what he did when they're older, even if they're too young to get it now. There's no shame in being an object of compassion.