The answer involves an immense diamond mine, a secret Utopian society on the island of Krakatoa immediately prior to its infamous explosion, many fantastic inventions, and a complicated restaurant-based economy. The inventions made me laugh, and you can tell du Bois enjoyed thinking them up - there is electrical furniture which raises up from the floor (and more importantly, sinks back to be flush with the floor, allowing easy cleaning), there is a bed with a continuous belt of sheet that rotates every morning so that a new section is on the bed and another section is washed in the basement, and there is a "balloon carousel" with twenty balloons connected in a circle, among many others.
In the end, a lot of fun! It would make a great read-aloud book (I'm saying that a lot lately, but it keeps being true!), and the author's illustrations fit the character of the story really well. They even *look* French (which the author was - I had to do a little bit of digging to figure out how this was eligible for a Newbery award, but it turns out he moved back to the US as a teenager).
An interesting tangent is the author's note at the front of the book, which I will quote here and allow you to enjoy in its entirety:
Just before publication of The Twenty-One Balloons, my publishers noted a strong resemblance between my book and a story by F. Scott Fitzgerald entitled "The Diamond as Big as the Ritz," published by Charles Scribner's Sons. I read this story immediately and discovered to my horror that it was not only quite similar as to general plot, but was altogether a collection of very similar ideas. This was the first I had heard of the F. Scott Fitzgerald story and I can only explain this embarrassing and, to me, maddening coincidence by a firm belief that the problem of making good use of the discovery of a fabulous amount of diamonds suggests but one obvious solution, which is secrecy. The fact that F. Scott Fitzgerald and I apparently would spend our billions in like ways right down to being dumped from bed into a bathtub is altogether, quite frankly, beyond my explanation. William Pene du Bois, 1947I haven't read Fitzgerald's story yet (though I do intend to!) but other reviews point out that although there are strong similarities on the face, the two stories' tones differ significantly. It does make one wonder, though, if there are certain things, both fantastic and fantastically mundane, that naturally come to mind when one thinks of diamonds and diamonds and diamonds...