In Chapter II of The Stranger, Meursault says:
I remembered it was a Sunday, and that put me off; I’ve never cared for Sundays.
In Chapter III, Meursault is prompted by Raymond to write a letter that may be used to lure his Arab (Moor) girlfriend back to him. Meursault responds:
I kept silence and he said it again. I didn’t care one way or the other, but as he seemed so set on it, I nodded and said, “Yes.”
In Chapter V, when asked by his boss if he'd prefer moving to Paris for a job promotion, Meursault responds:
I told him I was quite prepared to go; but really I didn’t care much one way or the other.
In Part II, Chapter II, Mersault finally starts to reflect on his life once in prison. He says:
THERE are some things of which I’ve never cared to talk. And, a few days after I’d been sent to prison, I decided that this phase of my life was one of them.
Later, in Part II, Chapter V, Meursault reflects on death:
And, on a wide view, I could see that it makes little difference whether one dies at the age of thirty or threescore and ten—since, in either case, other men and women will continue living, the world will go on as before.
So, Meursault doesn't care about that which most of the culture cares about: Sundays, revenge, job promotions, prison, and even death. As an absurd hero, Meursault is either in denial or in a state of repressed anger about the importance of these events. In short, Meursault hates death, loves life, and scorns the gods. He does not see any religious importance to the ritual of Sunday, the point of menial labor, or the fear of death--for he, like his mother, is ready to live his life over again just the way he lived it in the first place--with no guilt or regrets.
In general, he implies this everywhere. But his indifference is often indirect. One example is in the courtroom when he makes his statement. He says he didn’t intend on killing the Arab, but he blames the sun and the heat. The court finds this morbidly amusing, but for Meursault, this is the truth. His self-alienation from the world allowed him to ignore the rules of society and only react to his immediate/present situation. At that time, from his insular perspective, it was hot and that was all he cared about; therefore, indirectly he didn’t care about anything else.
Another big example of Meursault’s lack of caring was during the night sitting vigil with his mother’s body; and of course his nonchalant line that begins the novel. Along the lines of ‘Mother died today. Or maybe yesterday; I can’t be sure.”
Another moment is when he tells the chaplain that he doesn’t believe he’s committed a sin; he’s only been found guilty of a crime. Since Meursault doesn’t care for society’s rules, being found guilty of a crime doesn’t seem to be of much importance to him. Since (this is an assumption) Meursault doesn’t believe in God (or if he admits the possibility, he would say that humanity could not possibly grasp the mind of God and therefore, again, society’s rules are still manmade and fallible) – whichever the case, Meursault doesn’t care about anything the chaplain has to say; even when the chaplain mentions most people repent if even to hedge their bets and save their soul. Meursault sticks to his philosophy that this chaplain is no closer to God than anyone and that the societal rules are meaningless impositions.