Why did Atticus remove his coat, loosen his tie, and unbutton his vest?

Atticus wanted to diffuse the tension in the court room. He wanted to lessen the formality of his job as to connect better with the jury.

Despite the stifling heat of the courtroom, it was not a real factor in Atticus's decision to unbutton his collar and vest and remove his coat. It was a shocking sight for Jem and Scout, since Atticus

... never loosened a scrap of his clothing until he undressed at bedtime, and to Jem and me, it was the equivalent of him standing before us stark naked. ( Chapter 20 )

It was Atticus's way of stepping down from his position of authority and reducing himself to a common man, like the country folk dressed in their work clothes who served on the jury: He was appealing to their "humanity and morality."

His voice had lost its aridity, its detachment, and he was talking to the jury as if they were folks on the post office corner. ( Chapter 20 )

The remainder of his summation was a plea to the jury to accept Tom's version of the events, to make the unheard of leap of believing a black man's word over the Ewells'. Atticus displayed "another 'first' ":

... we had never seen him sweat--he was one of those men whose faces never perspired, but now it was shining tan. (Chapter 20 )

This was a case Atticus had "hoped to get through life without," and he knew he must take a different route when addressing the jury than he had ever attempted before.