What unusual nature by Atticus provoked the children's anxiety and concern?

I'm not exactly sure of what "unusual nature" you are referring to. Atticus does explain the concept of "rape" to Scout, much to the dismay of Aunt Alexandra, but I'm not sure that is what you are looking for. 

I mean to say his behaviour of Atticus

I think the threats of violence are increasing and Atticus, usually cautious,  is dismissing them. They are concerned that the "Sarum bunch" will cause some trouble, but Atticus thinks they won't do anything (such as a lynching) on a Sunday night. Mr. Link Deas warns Atticus that he has everything to lose from the trial, but Atticus says that he wants the truth to come out. Jem is concerned that the men outside mean Atticus harm, but Atticus assures him later that those men are his friends and are not part of a gang or the Ku Klux Klan, whom Atticus claims is gone and will never come back.