This question seems more related to social interactions than pure French language, but here are my two cents about it :

Tutoyer is informal, while vouvoyer is more formal. In this example, the fact that the magistrate switches to “tu” instead of “vous” would indeed indicate a willingness to get more “personnal”, closer to Meursault, as someone you know might be, on the contrary of someone using “vous”, implying some kind of distance. This also explains why Meursault contrasts it with “en avoir assez”, which seems to me to be perfectly transcripted in the translation.

Adding to Laurent’s answer, it may also be that someone switches to tu when high on emotions.

This actually happened to me recently when, during a formal but (friendly) hated discussion someone said tu parles ! “. This is not tutoiement but shows that some expressions may be influenced by emotions, yielding a short-term tutoiement which then switches back to vous

C’est exactement la réaction du juge qui sort de son rôle impartial et se laisse envahir par sa croyance qui lui ordonne de combattre les hérétiques, ou au moins de les ramener sous sa coupe, qui ne peut accepter que l’Autre vive dans un monde différent du sien.

Pour un juge démocratique, il se comporte comme un inquisiteur et va suivre les même procédure… jusqu’à la mort.

Meursault le pousse dans ses limites de raisonnement, dans son incapacité a accepter que les autres ne soient pas lui-même, il est “dos au mur”, il se rend compte que Meursault est aussi un homme, et il fait une manœuvre amoureuse : je cède un peu de moi, je te confie un secret, mais c’est pour mieux t’avoir à ma merci.

Ce tu est un cadeau (empoisonné) que l’homme juge fait à l’indigène (la scène se passe dans une colonie française) Meursault, en lui dévoilant un peu de lui-même, c’est une sorte de potlatch, il en attend en retour un cadeau encore plus grand : l’acceptation de Dieu comme seul maître.

Meursault, intègre dans ses pensées lui fait comprendre… qu’il ne l’a pas compris.

Like @laurentS said for the general meaning of the tutoiement.

I will clarify in this context of Camus: the sudden switching to “tu” is an attempt to get closer to convince the other, but it is in the same time an unsollicited entrance in some’s privacy.
The text clearly shows an act of despair / bullying, and it annoys Mersault who fakes to agree with the judge.
Then the judge express victory thinking he convinced Mersault, which he did not.

As rightly mentions user Laurent S., this question lies outside the bounds of linguistics and belongs properly to the domain of literary criticism, more specifically one of its elementary foundational bases, explication. Nevertheless, as many indulged in the exercise of trying to make some sense of this text, I’ll do so as well, hoping not to be the straw that breaks the camel’s back. No, there is not a basis for invoking any sincerety or solicitousness on the the magistrate’s part, there is even ground to discredit that inference as being very plausible, although not enough to replace it by anything convincing. Suggestive enough is user Laurent S.‘s remark to the effect that all eventualities are open as to the prime mover in the magistrate’s behaviour.

The interpretation of “mais” is different¹ as I think that there is not exactly an opposition but the signifying by “mais” of a refusal, that on the ground of the following acceptation of the word (TLFi) and the context;

En partic. [Le locuteur refuse la situation impliquée par la continuation du discours précédent et introd. une rupture dans ce discours (changement de thème, de point de vue, etc.)]

Meursault refuses the idea of the “tutoiement” and he should take some step in the sense of not allowing it without at least contesting it as characteristic of an unusual attitude; that’s his state of mind; however, he is tired of it all (“j’en avais assez”) and doesn’t want to react, he gives up his idea; it’s being told us next, he is moved by the wish of getting rid of his interlocutor, not of striving to either oppose or placate him, and so he feigns approbation. He is not aware then of this would be message of sympathy; on the contrary of the effect this would have, his aversion is enhanced.

The clues that might give away the magistrate’s attitude, his “mode” of behaviour, are quite revealing, archetypal, he fits into a pattern; what we can’t be sure of is whether he is really behaving so or putting it on somehow.

He insists on a religious theory, does so vehemently and boastfully, speaking his words too loudly, makes to Meursault unreasonnable, ridiculous demands for commitment to “his” salvation, not Meursault’s, having overall an attitude akin to that of an inquisitor¹, and finally he crowns it all by a most unequivocal act, the referring to Christ by the pronoun « celui-là »; it’s enough to conclude, in this particular context, that he is given to second-rate bar room histrionics. No one that is truly a christian can exhibit the attitude this implies: the use of this pronoun in such a vehement manner aims at imparting the notion of a certain mastery of the thing named (Christ), a certain exclusivity on the knowledge of it, and it aims at being threatening. There is no doubt that the effect desired by the magistrate, whatever the genuineness of the means, is to impress Meursault, more, to browbeat him, and the “tutoiement” is hardly a means to communicate friendly feelings in that case, it could be meant rather to do away with the distance the “vouvoiement” preserves so as to be more sanctimonious. I would hardly give a thought to the possibility of a carrot and stick approach.

All translations, “my poor young man”, “I noticed that his manner seemed genuinely solicitous when he said, “My poor young man”” and “I was struck by how sincere he was” miss the point in my opinion : you do not change from “vous” to “tu” in French out of compassion.

• “my poor Young man” — Say, buddy/mate, …
• “I was struck by how sincere he was” — I noticed him getting fresher,

¹ _{I arrived at that conclusion independently of its mention in the text of comments and answers, which I hadn’t read completely.}

I completely disagree with Laurent S. here:

the fact that the magistrate switches to “tu” instead of “vous” would
indeed indicate a willingness to get more “personnal”, closer to
Meursault

This is only true if the “tutoiement” goes in both direction, otherwise the effect is very different. As cl-r said, both locutors are fully aware that this “tutoiement” is a (crude) trap.

In this context, the “tutoiement” can indicates the will of the magistrate to infantilize his interlocutor: He is socially superior to Meursault, so he can “tutoyer”, while Meursault must continue to “vouvoyer” (as can a teacher with a student, or a boss with an worker). He add a distance between them: I’m right, you’re wrong, because I’m wiser (more educated), richer and so on.

It’s also a religious reference to “we are all children of God”. If you become my religious brother (Christian), you will gain the right to “tutoyer” me. Then you would become my equal (somehow). But Meursault do not try to compromise and remains faithful to his current paradigm of thought, he refuses to become a Christian. He does not want to listen more the arguments of a “religious fanatic” and prefers to continue to “vouvoyer” the magistrate, even if the magistrate “tutoie” him. Meursault maintain the distance between them.

J’ai bien remarqué qu’il me tutoyait, mais j’en avais assez.

He concedes to be inferior to the magistrate, because he wants to finish this masquarade quickly and arguing can’t improve the final judgment.

Il est retombé sur son fauteuil.

In the end the magistrate discovers that he failed to convince his interlocutor by threat of more severe punishment (as he is used to). It’s a shock to see a man ready to die without begging him. This magistrate is a caricature of arrogance.

I was struck by how sincere he seemed, but I had had enough.

Seems to me a weak translation (we lose the aspect: he takes me high), but conciseness is important here to reflect the brevity of the decision making (Meursault’s intransigence).

I noticed that his manner seemed genuinely solicitous when he said, “My poor young man”–but I was beginning to have enough of it.

Seems much closer of the double meaning of the French version:

• My poor young man for he takes me high.
• genuinely solicitous for he offer (amicably) a way out that usually the suspects accept.

But too complicated… That’s probably the oldest translation… (not in good way!)