... are classifications that have always mystified me.

The great author Pelham Grenville "Plum" Wodehouse uses these terms every so often in his books. They generally are used to refer to a person or a group of people, in the context of gentlemen's clubs (no, not that sort).

An example might go like this: "At the Drones Club, the Eggs were in a fine good mood, having broken the bank of every casino in the resort. In sharp contra-distinction, the Beans were downcast and the Crumpets were trying and failing to drown their sorrows in gin. The lone exception was Albert, a hardnosed and lucky Crumpet, who, due to an unforeseen turn of events, had resisted the temptor of the tables and fallen in love in the same evening."

So what do these mean? I have no idea, even though I've read probably at least half of Wodehouse's published works (of which there are many), including all books in the sagas about Jeeves and Wooster, Blandings Castle, and Psmith.

Does anyone know?

no idea, but as a guess id say its the authors terminology for the elite (eggs), merchant class (beans), and peasants(crumpets).
If i were to analyze it, i would say the eggs are considered elite because eggs are fragile and decadent.
the merchants are always counting their beans to make sure theres more coming than going...
Crumpets...well its a bread reference, probably coming from times where the poor peoples diet was almost entirely of bread. although a crumpet is a dessert bread, so perhaps its to say the poor arent so bad off?