"Maman died today. Or yesterday maybe, I don't know."
This could be explained as Meursault merely being stuck with grief and unable to grasp his mother's death. However throughout the novel we can see Meursault weaving his way through life with the same indifference toward other people and toward life itself. Unless something directly affects him, Meursault takes very little interest in it. His reaction to his mother's death and his unemotional state at her funeral will later be used to condemn him at the end of the novel.
The Stanger takes place in sunny French-Algeria, pressed up against the Mediterranean Sea. Meursault attends his mother's funeral, returns home, relaxes at the beach and has casual sex with a girl he knows. What he remembers most about the funeral was being hot and uncomforable. This is perhaps what stands out most about Meursault, his need to satisfy his immediate physical requirements. If he is hungry he eats, if he wants to have sex he can find a girl, he smokes when he wants and finds pleasure in idle amusement. Meursault is not the kind of guy who would have a 401k plan, he does not think about long-term happiness or consequences, only immediate satisfaction for himself.
Meursault still has friends, holds down a decent job and the girl he slept with wants to marry him. However his apathy is apparent in all these situations. For example, his response to the girl is that it does not matter to him if he gets married, but if it makes her happy it it OK with him that they get married. When Meursault is offered the chance to relocate to Paris for work he does not seem to care, except that he would miss the sun and the sea.
Meursault is invited by one of his friends, Raymond, to spend some time at a beach house. While Meursault seems cool-headed and almost passionless, Raymond is the opposite, dressing in flashy clothes, having a violent temper and a loud personality. Raymond has upset some Arabs who are also staying on the beach and one of the Arabs cuts Raymond. Seeking to perhaps settle the score Raymond brings a gun with him to the beach, but even-keeled Meursault persuades Raymond to give him the gun. Meursault later finds himself alone on the beach when he confronts one of the Arabs, whom he kills with the gun.
The second half of the book focuses on Meursault's trial for murder. Most people seem to think that Meursault's trial will be dismissed as manslaughter, especially as another high profile case for patricide is making its way through the courts. However, when people learn of Meursault's shocking indifference at his mother's funeral he becomes regarded as some kind of sociopath, devoid of feeling for his fellow man. Meursault's conviction for murder becomes more about his apathy at his mother's funeral than for the killing the Arab on the beach.
There is no difference between dying now or 20 years from now. People won't even know that he has died or even lived today. It reminds me of my wedding day. It is one of the most important days in my life, but to almost everyone else it is just another Saturday. The tender indifference of the world.