What does atticus do that causes scout to say that she felt he was the "bravest man that ever lived"?

Mrs. Dubose is a deeply unpleasant old lady who lives down the street from the Finches. Scout describes her as "plain hell." It's not surprising she feels this way when you consider how thoroughly mean the old lady is to everyone. Whenever Scout and Jem walk past her house she scowls at them and hurls abuse. She is a pretty scary character; and that's not to mention the old Confederate pistol she's alleged to keep under her shawl.
As expected, Atticus shows much greater wisdom and empathy. Each time he passes the Dubose residence, he politely doffs his hat and wishes the old lady a cheery good morning. This impresses Scout so much that she describes her father as the bravest man who ever lived. Ironically, Atticus thinks that Mrs. Dubose is the bravest person he's ever seen. She is a morphine addict, determined to quit her destructive habit before she dies. Sadly for her, she doesn't manage to achieve her goal. Yet Atticus admires her courage nonetheless:

I wanted you to see what real courage is, instead of getting the idea that courage is a man with a gun in his hand. It’s when you know that you’re licked before you begin but you begin anyway and you see it through no matter what. You rarely win, but sometimes you do. Mrs. Dubose won, all ninety-eight pounds of her. According to her views, she died beholden to nothing and nobody. She was the bravest person I ever knew.