The short answer is that there is no reason behind Meursault killing the Arab man. Meursault is not acting in self-defense, since he is not afraid, even when the Arab man takes a knife out. He is not acting out of malice, either. While the heat on the beach is oppressive and making Meursault miserable, and the sun is glaring in his eyes, that, too, is an insufficient reason for his choice to kill a man.

In fact, everything about the murder seems random and based on chance. The Arab man whom Meursault shoots is the brother of Raymond's mistress—these are both people whom Meursault is indifferent toward. He only comes to possess the murder weapon by chance (he took it away from Raymond to prevent him from killing the Arabs out of anger earlier in the chapter). The sun just happens to be oppressive and unbearable that day.

That the murder seems senseless and random is intentional. Camus provides Meursault with no motive because Meursault is meant to be a human reflection of the nature of the universe as Camus sees it: indifferent and irrational. Any attempt to find a proper reason behind the killing on the part of the reader or the other characters in the narrative is doomed to failure because the killing is meant to be absurd.