How weird is it that the whole story is told from Mrs. Frankweiler's perspective, even though she's not involved in most of it?
At first glance (if you ignore the introduction), you might think that this is a straightforward, third-person omniscient narrator, someone who can see everything that Claudia and Jamie are doing (and thinking) but is detached from the story. But then we get all these odd asides to Saxonberg, and the occasional "I" thrown in there.
For example, the story is just plodding along and then the narrator wants to talk about her connections to folks at the museum: "My chauffeur, Sheldon, has a friend named Morris who is a guard at the Metropolitan" (3.20). Huh? Who is this Sheldon character and who is his friend Morris? That's weird.
And then the narrator scolds this Saxonberg person for not being into art museums enough: "You've missed all this, Saxonberg. Shame on you! You've never set your well-polished shoe inside that museum" (3.22). We're starting to realize that Mrs. Frankweiler knows quite a bit about what the kids think and do, and that there's a reason she's telling all of this to Saxonberg (whoever he may be).
It's bizarre enough to have such a peripheral narrator (when Claudia or Jamie could surely do the job). But it's even weirder to be telling it all to someone who wasn't even there—or, for all accounts, wasn't even in the story. The choice to tell the story to Saxonberg adds another layer of mystery to the story.