Often, the way that people feel about an issue is different than the way that they behave about an issue. For example, consider the Tom Robinson case. He was found guilty, which means that the people on the jury voted him as guilty; however, most of those men knew in their hearts that Tom Robinson was innocent. Atticus helped to prove pretty clearly that Bob Ewell was the true criminal in the case, and Tom was the unlucky fellow that got pinned with the crime. So, in "the secret courts of men's hearts," they knew that Tom was innocent. However, knowing that and acting on it are two entirely different things. To act on that truth and give him an innocent verdict, would be to make a monumental and difficult stand in the face of generations of racism, prejudice, fear and intolerance. If they would have given Tom an innocent verdict, they would have been spat upon, just like Atticus was. Their families and lives would have been threatened. This doesn't justify the guilty verdict, but it helps explain it a bit. It takes a very long time to change public sentiment, and to change behavior; changing that behavior starts in the hearts of men. Atticus was satisfied to a certain degree about the outcome of the trial, because he never fully expected to win. But, noting how long it took the jury to come up with their conviction, he was convinced that he had changed some of the hearts of the men on that jury so much that they had tried to argue for Tom. Tried and failed, granted, but at least they tried, and as Atticus put it, that was "the shadow of a beginning."
I hope that those thoughts help a bit; good luck!