How does Meursault explain his nature in The Stranger?
He doesn't explain his nature because he doesn't have a human nature. He lives his life on his own terms, not according to any nature. That's the point of the existentialism and absurdism. Camus' philosophy says that Meursault is born first and defined later; he is not born with a predetermined or socially determined human nature.
Camus champions Muersault to be his absurd hero: to love life, to hate death, and to scorn the gods. Meursault stands by his choices throughout the novel: he doesn't regret not crying at his mother's funeral ("No one and I said no one had the right to cry for her"); he doesn't regret killing the Arab; and he doesn't regret or feel guilty about telling the Magistrate that he doesn't believe in God. Having to answer to society's judgments is an act of conformity.
Meursault does not defend himself in court. In fact, he goes to his death in hopes that a crowd of people will jeer at him as he is guillotined. He would have lived his life over exactly the same way, with no regrets. The absurd hero is defined by his actions, not so much his words. Having to explain one's nature undermines the freedom of choice in the first place, according to Camus. Man is condemned to be free, and a human nature limits one's choices.