Atticus, a very polite, formal man who was a stickler for manners, was never seen without tidy clothing; he was always buttoned, his tie was up tight, and no one had ever seen him any differently, including his children. But in the courtroom, as Atticus is about to give his closing testimony, he does something unusual. He "unbuttoned his vest, unbuttoned his collar, loosened his tie, and took off his coat." Scout says of this that that she "never saw him do [that] before or since, in public or private...he never loosened a scrap of his clothing until he undressed at bedtime." This just emphasizes how important Atticus felt his closing testimony was; he wanted to appear like the other men. He wanted to seem approachable, honest, vulnerable and sincere, and loosening his tie was a way to say to the jury, "Look here. I'm going to be honest with you for a bit. Let's have a chat, just you and me." It made him seem more down-to-earth, and one of the gang. It is an interesting strategy, or maybe just a natural instinct that went in line with Atticus' sincerity in the moment. But, according to Scout, it was highly unusual.
When Atticus is giving his final summation in Tom Robinson's trial, he does something that Jem and Scout have never seen him do in public before: "he unbuttoned his vest, unbuttoned his collar, loosened his tie, and took off his coat." Scout says that Atticus's state of undress is "the equivalent of him standing before us stark naked." Scout and Jem are aghast at Atticus's unwonted familiarity, and Scout notes that when Atticus addresses the jury, his voice is more familiar than it usually is. She notes that his voice "had lost its aridity, its detachment." Atticus is speaking to the members of the jury as if they were neighbors he had bumped into on the street corner, and his voice and state of dress suggest that he wants the jury to see him as one of them. He doesn't want to seem erudite or detached; instead, he wants the jury to see him as a fellow member of the community. If they do, Atticus thinks, the jury might consider that the claims against Tom Robinson are false.