In To Kill a Mockingbird, what does Scout mean when she says, "in the secret courts of men's hearts, Atticus had no case"?
Scout is observing how Atticus, a reasonable and empathetic man, cannot hope to prevail with racist men and their ignorant, stubborn beliefs. This is a theme that is developed in more detail following the trial. In chapter 23 Jem is frustrated by the jury's refusal to dismiss the charges against Tom, even though he was clearly innocent. Atticus knew this would happen and tried to prepare Scout and Jem for it. That is why he had to walk a fine line - and that is one reason for having Jem read to Miss Dubose - Atticus wanted the kids to realize that why Maycomb's residents might be stubborn and ignorant in their beliefs, they were still their neighbors and friends.
It means that in their hearts, these white southern men had already made their decisions. There was no way they would rule in favor of a black man against a white woman, no matter how much evidence Atticus put in front of their faces proving Tom Robinson's innocence. At that time in history, it just was not going to happen.