Initially, there isn't much. The final words of Part I read this:
I wanted to hear the murmur of its water again, to escape from the sum and the effort and the women's tears, and to relax in the shade again. But when I got nearer, I saw that Raymond's Arab had come back. He was alone. He was lying on his back, with his hands behind his head ... As far as I was concerned, it was all settled and I'd gone there without even thinking about it. As soon as he saw me, he sat up lightly and put his hand in his pocket.
There was a bit of a defensive mood at this point in the story. You have to remember that Mersault had simply gone with his friends for a day on the beach. Two Arabs were seemingly following them. Prior to these paragraphs, these Arabs had already engaged in a bit of a fight with Mersault and his friends, with one of them having slashed his friend with a knife. Moving on... (emphasis mine)
I took a few steps towards the spring. The Arab didn't move. Even now, he was still some distance away... I waited. The sun was beginning to burn on my cheeks, and I felt drops of sweat gathering in my eyebrows. It was the same sun as on the day of my mother's funeral, and again it was my forehead that was hurting me most and all the veins were throbbing at once beneath the skin... I took a step, just one step forward. And this time, without sitting up, the Arab drew his knife and held it out towards me in the sun... All I could feel were the cymbals the sun was clashing against my forehead and, indistinctly, the dazzling spear still leaping up off the knife in front of me. It was like a red-hot blade gnawing at my eyelashes and gouging out my stinging eyes.
This passage is also interesting. Mersault was undoubtedly under the sun, and it was affecting him. The parallel between the sun and his mother's funeral is also somewhat significant. At his mother's funeral, he was largely alone and almost kept to himself. He also defied many societal standards related to his mother's funeral.
The passages in these pages offer that parallel. Mersault wants to keep the peace within him, and he doesn't want anyone around him. The presence of the Arab disturbed him. When the Arab drew his knife, it disturbed this peace, bringing about provoking actions:
... That was when everything shook... The sky seemed to be splitting from end to end and raining down sheets of flame. My whole being went tense and I tightened my grip on the gun. The trigger gave, I felt the underside of the polished butt and it was there, in that sharp but deafening noise, that it all started. I realized that I'd destroyed the balance of the day and the perfect silence of this beach where I'd been happy. And I fired four more times at a lifeless body and the bullets sank without leaving a mark. And it was like giving four sharp knocks at the door of unhappiness.
These passages provide a direct narrative for what directly instigated the act. Mersault may not have been completely conscious of his act.
But what I find interesting is the last sentence. You see, this novel is largely about existentialism. Albert Camus, the author, was an existentialist. The murder came as a shock - here we are looking at the life of an absurd man. He doesn't necessarily follow social norms, he's not married... etc. Earlier in the chapter, he said:
When Raymond handed me his gun, the sun glinted off it... I realized at that point that you could either shoot or not shoot.
You can shoot or not shoot. It doesn't matter. If someone is murdered or not, the world doesn't care. The theme of existence is largely at play. Whatever happens, it doesn't matter.
No matter how you look at it, there is not a reasonable explanation for Mersault's actions. It conveys the theme of existentialism, as well as the title of the book overall - Mersualt is an outsider in the way that he defies the laws of society. You see this most notably in his trial, where he shows remorse, and is ultimately convicted not because he committed the murder, but because he doesn't live up to what society wishes.