What does Atticus mean when he says "don't you worry about anything" in To Kill a Mockingbird?

When Atticus tells Scout it is not time to worry, he means that the trial will be difficult but they are safe now.

Atticus’s sister and Scout’s aunt, Alexandra, asks him to impart upon his children the “facts of life” ahead of the trial.  Basically, she wants him to explain the social hierarchy of Maycomb and their place in it.  The Finches are superior to others because they are from a good, old propertied family. 

When Atticus tries to explain this to them, all it does is worry Scout. This is not the father she knows.  Her father doesn’t care about social class, or think that some people are better than others.  If he is talking like this, it means that things are serious and she doesn’t understand why. 

Atticus scolds her for fidgeting with a comb, and that is all Scout can take.  She begins to cry, feeling overwhelmed by the gravity of the situation.  Atticus tries to reassure her that things are still all right.

“You really want us to do all that? I can’t remember everything Finches are supposed to do…”

“I don’t want you to remember it. Forget it.” (Ch. 13)

At this point, the town is gearing up for the trial.  Things are getting tougher and tougher for the Finches, and of course for Atticus.  He has taken the case because he was told to, but the whole town seems set against him for it.  Scout is too young to understand why people seem to hate her father all of the sudden.

Jem is more aware of what is going on than Scout.

“It’s this Tom Robinson case that’s worryin‘ him to death—”

I said Atticus didn’t worry about anything. Besides, the case never bothered us except about once a week and then it didn’t last. (Ch. 14)

Jem understands the situation better because he is older.  He tells Scout she doesn’t have the attention span to appreciate what is going on, and he is partly right.  If the situation does not present itself to her, Scout doesn’t think about it.  It is only when someone insults her father that she concerns herself with the Robinson case.  Jem understands the impact it is having on their lives.