The story published as "The Night the Bed Fell" in the New Yorker is, in fact, the first chapter of author James Thurber's memoir, My Life and Hard Times, published that same year (1933). Dorothy Parker called the book a work of (perhaps mad) genius, and humorist Ogden Dash characterized it as "just about the best thing I ever read." Thurber introduces this chapter as an anecdote from his childhood. He repeats it often among friends, he says, emoting hysterically to perform it most accurately and entertainingly. He also notes his friends have grown weary of hearing it. In comic fashion, then, Thurber informs his readers that this will be a story of remembrance. Like most such stories, it has a beginning, middle and end. It's a story from Thurber's life, so of course he recalls it in the chronological order in which he first experienced it.
Thurber sets the stage by describing his extended family's sleeping arrangements, building interest by relating an eccentric fact about each person in the story, especially their respective phobias. The writer claims he's "straying" from the narrative, but in fact he's using these preceding details to build our ability to visualize and thus take interest in the story he's about to relate. Thurber refers also to his temporarily absent grandfather, a seemingly irrelevant detail that will come into play at the end of the story. Meanwhile, the short sentence "By midnight we were all in bed" lets us know the stage is set and we're about to dive in fully.
Note that by using the phonetic exclamations "Ugf, ahfg!" and "Gugh," Thurber invites us to imagine himself yelling and gesticulating as if he were telling us the story in person. He opened that possibility by describing his own storytelling techniques before he began the narrative proper.
Thurber's phrase "I found at last the light switch" is a temporal cue to readers that he's wrapping up his story. The comic debacle is about to be resolved, with a possible thematic conclusion and/or narrative denouement to follow. Sure enough, the family dog bounds in for one final comic complication. All parties are calmed and their partial understanding of the hubbub resolved as the matter is "finally put together like a gigantic jigsaw puzzle." In the story's concluding line, Thurber ends the chronological narrative but also brings his structure full circle by allowing his mother to refer once again to the absent grandfather. We can almost imagine Thurber's in-person audience laughing and clapping as he takes a raconteurial bow.