Atticus means that when something like the conviction of Tom Robinson occurs, it is only the children who weep, because only they will recognize and admit the utter injustice of it.
The entire quote in Chapter 22 is "They've done it before and they did it tonight and they'll do it again and when they do it - seems that only children weep". Atticus tiredly says this when he returns home immediately after he has lost his quest to acquit Robinson. It is obvious that Robinson has been declared guilty simply because he is black, and Jem cries angry tears over the wrongful conviction. Atticus is at a loss to reply to Jem's question, "How could they do it, how could they?" All he knows is that adults continue to perpetuate the system of inequality and oppression with all kinds of twisted justifications, refusing to accept the simple fact that it is neither right nor fair. It is only the children who see clearly to the truth, and weep.
On a literal level, the line refers to the fact that it seems like only the children (mostly his children) were surprised or emotionally moved by the trial's verdict. They are still innocent, and so can still be hurt by the world's injustices. Adults, on the other hand, are armored and cynical, and so don't cry just because something bad happens. Now, there are two more ambitious ways this can be read. One is within the book, linking the line to the novel's epigraph. To be a lawyer is to put aside one's childhood heart: to armor up.
More generally, Jesus spoke of becoming like a child again, and that sort of open-hearted compassion would be familiar to the citizens of Maycomb from their churches.