In this quote, Scout wants to highlight how difficult it was for a black man to get access to justice at the time. She tells us that Tom Robinson had been given "due process of law." This implies that Tom Robinson was treated fairly in the eyes of the law because he had had a lawyer and a trial.
However, Scout also highlights the fact that, besides these advantages, Robinson's trial was definitely not fair and just. In fact, he was harshly judged because of his skin colour, which many people took to be almost evidence of Robinson's guilt. Therefore, as Scout highlights, Tom had "lost" in front of his peers, and he was never able to gain support from people who were against him from the beginning. Tom had lost the case in the "secret court of men's heart."