There are two main approaches to finding an explanation to the killing of the Arab. One is based on the novel's meaning (i.e. looking beyond character analysis); another approach looks beyond over character motivation to find motives that are not apparent at the surface.
What does not work well is looking for motivation at the surface level of the text. This is due to Camus's narrative technique. The entire story is told from a first-person perspective, i.e. a type of perspective that, in French literature before The Stranger, had been used to allow introspection. But introspection is exactly what we don't get; instead we get a narrative that gives us roughly the same insight into Meursault as a third-person perspective, almost as if an outsider were observing Meursault; only it is Meursault himself telling that story. Several early critics have commented on this, for example Jean-Paul Sartre, Maurice Blanchot and Claude-Edmonde Magny.
For this reason, analysing Meursault like a character from a realist novel is a vain pursuit. Camus himself remarked in his notebooks (in a draft for a letter in response to a review of The Stranger), "Realism is a meaningless word (Madame Bovary and Demons are realist novels and they have nothing in common." ("Le réalisme est un mot vide de sens (Mme Bovary et Les Possédés sont des romans réalistes et ils n'ont rien en commun." Quoted in Bernard Pingaud: L'Étranger d'Albert Camus. Gallimard, 1992, page 187.)
As mentioned above, Meursault tells his story with a kind of outsider's objectivity. He is also presented as a character that refuses to lie and play by society's conventional rules. After his mother's death, he does not seem to be mourning: he is not interested in having his mother's coffin opened so he can take a last look at her, he smokes during the wake, even falls asleep, and he doesn't weep during the funeral. These are examples of not playing by the normal social rules: he does not want to feign grief when he doesn't feel any. His lawyer later tells him that this could be used against him during the murder trial and suggest that he might say that his insensitivity at the funeral was not real but the result of an attempt to control his emotions. Meursault immediately rejects the idea because it was "untrue". This illustrates his refusal to lie.
Later on, during the trial, the killing of the Arab becomes an issue of secondary importance compared to Meursault's insensitiveness. Witnesses are questioned about Meursault's character and about his behaviour during and after the funeral but not about what they know about the killing. Meursualt is accused of "morally killing" his own mother; during his closing speech, the prosecutor even claims Meursault is also guilty of killing his father. (In reality, the cause of his mother's death is not known but she was definitely not killed by Meursault, while his father died when he was still a small child.) The prosecution's closing speech appears to mention the killing of the Arab only briefly, adding that Meursault "knows the value of words", which is another allusion to Meursault's commitment to truth (though probably not from the prosecutor's point of view).
From this point of view, as I already mentioned in my discussion of the Czechoslovak traveller's story, Meursault is put on trial for refusing to lie and not playing by society's conventional rules. However, a novel about a character that is prosecuted solely for this reason would be implausible. This is why society needs an excuse to put Meursault on trial, and by letting Meursault unintentionally kill an Arab, the novelist provides exactly that excuse.
Another way of approaching Meursault's "motivation" to kill the Arab is provided by psychoanalysis. As mentioned before, Meursault does not show any signs of grief or mourning after his mother's death. From a Freudian point of view, a person in mourning can feel their loss in an external way. In that case, they can reform their feeling of loss and the process of mourning ends with a kind of acceptance. Or a person in mourning feels their pain in an internal way; it is felt in the unconscious only, where the person is not aware of it. In that case, according to Freud, the grief is so heavy that it is repressed and cannot be processed by the conscious mind. This would explain why, at the funeral, Meursault behaves in a way that witnesses later describe as insensitive. It would also explain why his mother is mentioned again and again later in the book, for example, when he hears Salamano weep over the loss of his dog, he suddenly thinks of his mother but can't explain why (end of Part One, chapter III).
Freudian interpretations take this further to explain other aspects of the novel. (See for example, Bernard Pingaud: L'Étranger d'Albert Camus. Gallimard, 1992; page 114-123.) At the end of a normal mourning process, the mourner eventually prefers himself over the lost "object"; at the end of a pathological process, the mourner is unable to give up the lost "object", ascribe the lost to himself and decides to share its fate (see Freud's Mourning and Melancholia). This would explain Meursault's unusual behaviour each time people ask him something about his mother. (For example, during his first conversation with the investigating magistrate in Part Two, Chapter I, the magistrate asks him whether he had loved his mother. Meursault answers, "Yes, like everybody", after which he notices that the clerk behind him suddenly interrupts his regular typing as if had mistyped something and needed to delete it.)
Shortly before that he had already thought to himself: "I probably loved mother, but that didn't mean anything. All sane beings had more or less wished for the death of those they loved." ("Sans doute, j'aimais maman, mais cela ne voulait rien dire. Tous les être sains avaient plus ou moins souhaité la mort de ceux qu'ils aimaient.") For Freudian critics, Meursault's wish for his mother's death is as if he had effectively killed here, so it would only be just for him to share her fate. From this point of view, the killing of the Arab is a roundabout way of committing suicide, using the justice system as an instrument. (This reading puts the prosecutor's claim that Meursault had "morally killed" his mother in a different light than the previous interpretation.)
Both of these interpretations are rather unintuitive but they uncover aspects of the novel that remain hidden when reading the novel and analysing its characters as if The Stranger were a realist novel.