Atticus has no case because of the racial prejudice toward blacks during that period of time, especially in the South. The "secret courts of men's hearts" refers to how a white person felt about a black person in his/her heart even if he/she didn't come right out and say they were prejudiced. Atticus presents an excellent defense for Tom Robinson, but he can't win because Tom is black. Had Tom been a white man, he probably would have been found innocent. This quote from the book captures the theme not only of racial inequality, but of conformity as well. Boo Radley is tortured because he is different also.

Ah, a good but sad question. She meant that no matter how well Atticus made his case in a court of law, he was going to lose. Why? Because of the attitudes in the town. Because "in the secret courts of men's hearts," they were biased, and would not allow a case with a racial split like this one (Black male defendant, white female accuser) to be resolved fairly.
Atticus took the case knowing he would lose, but with the hope of creating greater fairness in the long run.
Greg

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.