Atticus is trying to demonstrate that Bob Ewell is left handed because Mayella’s injuries are on the right side of her face, showing that he is the one that hit her and not Tom.
During Tom Robinson’s rape trial, one of the first things that Atticus tries to establish is that Mayella Ewell was not raped at all. She was beaten up, but not raped. He also wants to show that her injuries were consistent with a left-handed person hitting her, because Tom Robinson could not have done it.
Atticus was trying to show … that Mr. Ewell could have beaten up Mayella. … If her right eye was blacked and she was beaten mostly on the right side of the face, it would tend to show that a left-handed person did it. (ch 17)
In fact, Atticus blatantly demonstrates that Bob Ewell is left-handed, the injuries on the right side of Mayella’s face were put there by someone left-handed, and Tom Robinson cannot use his left hand.
[Bob Ewell] swore out a warrant, no doubt signing it with his left hand, and Tom Robinson now sits before you, having taken the oath with the only good hand he possesses- his right hand. (ch 20)
This line of questioning is designed to bring into doubt the fact that the crime took place and the fact that Tom Robinson could have done it. Atticus wants to show that Tom Robinson did not attack Mayella, but her father did. He beat her when he caught her with a black man.
Unfortunately, despite the evidence to the contrary, the jury convicts Tom Robinson. This is because they are so steeped in racism that it is impossible for them to take the word of a black man over that of a white woman.