The story is told from the point of view of Mrs. Frankweiler, who is both perfectly matter-of-fact and very old (82 years old, to be exact). With all those years under her undoubtedly expensive belt, she tells the story from the perspective of an older, wiser person who knows all the details and can fill you in on them. Claudia and Jamie's story is told in great detail, with clarity and conciseness.
For example, this is how she opens up the story once she's decided to tell it all to Saxonberg:
Claudia knew that she could never pull off the old-fashioned kind of running away. That is, running away in the heat of anger with a knapsack on her back. She didn't like discomfort; even picnics were untidy and inconvenient: all those insects and sun melting the icing on the cupcakes. (1.1)
She's able to set up the story perfectly, including both the premise and an introduction to Claudia and her personality. If we didn't know any better, we'd think that a professional writer wrote this—not an old woman with a pen and an extremely long letter to write.
The only points where Mrs. Frankweiler interjects are when she feels it is necessary to impart some of her old lady nuggets of wisdom. She wants Saxonberg to know about all the awesome things in the museum, and the tidbits of facts that she's privy to:
Now, Saxonberg, I must tell you about that Egyptian tomb called a mastaba. It is not a whole one; it is the beginning of one. You can walk in it. (7.27)
What a smart old lady. She's certainly got some nuggets of factoids on her. When bringing up Michelangelo, she also takes a moment to lecture Saxonberg again on the magic of learning about old famous artists. She's quite a lady, that Mrs. Frankweiler. She may be able to tell a mean story, but don't expect her to keep her opinions to herself.