The great basis of comedy is hyperbole or exaggeration. If understatement is the great tool of tragedy—the author or narrator getting out of the way so that readers can experience painful emotions for themselves—going over the top into outrageousness is what draws laughs out of people. If a person stumbling on a step might raise a laugh, a person tumbling down a flight of steps—as long as he bounces up unhurt—is worth a dozen laughs.
Wodehouse was a master of verbal hyperbole as way to draw out laughs and highlight the ridiculous. One example from the story comes from Lord Emsworth's distress at having a second son, followed by the idea that second sons (who won't inherit the estate) are not popular with aristocrats. Lord Emsworth's irritation with his second son is contrasted absurdly to the male codfish loving all of its millions of children:
Unlike the male codfish, which, suddenly finding itself the parent of three million five hundred thousand little codfish, cheerfully resolves to love them all, the British aristocracy is apt to look with a somewhat jaundiced eye on its younger sons ...
It is hard not to laugh at Lord Emsworth being compared to a codfish.
Another very characteristic way Wodehouse makes us laugh is through allusion to other works or traditions of literature. He assumes a shared educational background in his reader. Therefore, when he compares the hapless and less-than-poetic Freddie to the exalted vision of Greek shepherds from classical pastoral literature, the disjunction raises a laugh:
into the range of his vision there came the Hon. Freddie. White and shining, he tripped along over the turf like a Theocritan shepherd hastening to keep an appointment with a nymph
It's equally humorous when Wodehouse compares the moronic Freddie to Hamlet:
Hamlet's society at Elsinore must have had much the same effect on his stepfather as did that of Freddie Threepwood at Blandings on Lord Emsworth.
Wodehouse's use of allusion can also be dry, as when he uses a familiar quote about news of the birth of Christ to comment on Freddie's news of being engaged to an American girl named Niagara:
it scarcely fell into the class of tidings of great joy.
Finally, another classic device Wodehouse uses is social reversal, sometimes referred to as the carnivalesque. In this instance, Lord Emsworth, the grand and wealthy aristocratic, is the imbecile who can't think to take the cap off his telescope while his gardener is all competence.
It's important to note that while Wodehouse specializes in creating upperclass imbeciles, he does so kindly—they are generally good, well meaning people. Wodehouse can make us laugh, but he does so gently.