The "stream of consciousness" in Atonement refers to its narrative mode. This narrative mode consists on random thoughts, memories, and in the description of events that are told in a disorganized way assuming that the narrator, which may or may not be the main character, is actively remembering them as they come up throughout the story. Another aspect of this narrative style is that the narrator constantly analyzes emotions and thoughts.
An example of this would be the following excerpt
Briony carried her half-smoked cigarette to the sink. She was feeling sick... Her sister's confirmation of her crime was terrible to hear. But the perspective was unfamiliar. Weak, stupid, confused, cowardly, evasive- she had hated herself for everything she had been, but she had never thought of herself as a liar
Atonement features a great example of the stream of consciousness narrative style because the events are told from Briony's point of view, as well as she can remember them, or as well as she can understand them; even, as she plans to accept or deny them. This is because, as a thirteen year old girl, Briony's testimony is based on her own perspective. As a young woman she has to explain to herself (as well as to the reader) her role in the events that led to the chaos in the Tallis family. Hence, the need to atone for her actions.
Her narrative combines memories, self-denial, opinions, realities, factual, and fictional events that only Briony can confirm. Therefore, since the story is told in this atypical, purely subjective, and disorganized manner, the narrative is deemed as a "stream of consciousness".
As into the sunset we sail. An unhappy inversion. It occurs to me that I have not traveled so very far after all... Or rather, I've made a huge digression and doubled back to my starting place. It is only in this last version that my lovers end well, standing side by side on a South London pavement as I walk away.
This particular example above shows the disorganized and sudden thoughts that come up as the story unravels, and how they express the subconscious of the narrator.