Scout's words refer to the incident in front of the jail, when Mr. Cunningham and his band of malcontents show up to lynch Tom Robinson.
Despite his calm manner, Atticus is cognizant of the danger he is in. When the men show up, his demeanor is steady and his mannerisms deceptively relaxed. It is obvious that he is bracing himself for an unpleasant situation and is aware that the Tom Robinson trial has angered many of his white neighbors.
We saw Atticus look up from his newspaper. He closed it, folded it deliberately, dropped it in his lap, and pushed his hat to the back of his head. He seemed to be expecting them.
However, Scout unwittingly becomes the catalyst for dispersing the tensions of the moment when she speaks to Mr. Cunningham. Her innocence and beguiling trust in her neighbor disarms the farmer. In the end, Mr. Cunningham orders the men to leave. Scout doesn't realize the full import of her actions until she lays in bed that night.
I was very tired, and was drifting into sleep when the memory of Atticus calmly folding his newspaper and pushing back his hat became Atticus standing in the middle of an empty waiting street, pushing up his glasses. The full meaning of the night’s events hit me and I began crying.
Although Jem instinctively realizes that his father was in danger earlier, Scout doesn't come to this realization until later. In her mind, the images of her father standing as a lone man against so many others is a disturbing one. This is why she bursts out crying. The author juxtaposes the imagery of Atticus' ordinary demeanor, with his glasses, newspaper, and hat, against the malevolence exhibited by the 'sullen-looking, sleepy-eyed men.'
It is with this image of a lone, heroic Atticus standing against the tide of public opinion that Scout comes to understand the seriousness of the Tom Robinson case. To her grief, she now knows that her neighbors are willing to resort to violence to protect their idea of true justice.