You would have to read The Fall in the original French to see what we’re talking about when we call it "formal." The text reads like a rather stiff, linguistically proper confession, making use of conditionals and subjunctives, as Jean-Baptiste comments himself. Remember when he gets all excited about you recognizing his imperfect subjunctive use?
Since we (and probably you, too) are dealing with translated works, the style you see is going to depend a little bit on the edition you use. We’ve read a few different English translations of The Fall, and none of them are overly-ornate as far as the actual language goes. In the French text, Jean-Baptiste addressed you with the formal you, vous, and not the informal, tu. This is to be expected in polite company, but the narrator continues to use correct and polite language throughout his story.
Jean-Baptiste even says, "For the statue to stand bare, the fine speeches must take flight like pigeons" [3.32]. The words themselves don’t matter, only the truth that lies beneath them. Given this, it’s not the word-choice that’s worth talking about, but the way that The Fall is structured and the way in which Jean-Baptiste executes his "confession." You have to admit, that the project itself is impressive: Jean-Baptiste draws you in under the pretense of talking about himself, slowly gets you increasingly invested in his tale, and finally reveals that he’s been talking about you this whole time. When you add in the fact that Camus is using Jean-Baptiste to make a point to you (the reader) by having Jean-Baptiste use you (the fictional character) to make his (Camus’s) own point, it’s downright brilliant.
Now, Camus did claim that technique is not more important than content in The Fall. In other words, don’t get so carried away with his brilliant execution that you miss the point of his novel.