The most impressive thing about "There Will Come Soft Rains" is that the author makes it look easy. The story is essentially characterless and conversation-less, and, with one minor exception, it all takes place inside one house. Making a story that compelling and engaging without common and reliable elements, such as any two characters talking or changing the location of some scenes, is a very difficult task.
Ray Bradbury finds ways to make his short story entertaining by decorating the language of the story with elements like rhyming, repetition, and alliteration. The effects of all of these are slightly different, but they all serve the same purpose of enhancing the story. Alliteration in this story certainly makes it flow more easily. It is easier for the reader's eyes to glide along long sentences if alliteration is used. Think of alliteration as a river that carries the audience through the story's twists and turns.
Using alliteration can be tricky, and the level that Bradbury employs in "There Will Come Soft Rains" would be excessive in almost any other story. Because of the tone and relative shortness of the story, however, the heavy use of alliteration adds rather than detracts. The alliteration makes this short story sound lyrical, almost like a children's tale that could have been read to the children that lived in that automated nursery.
At times, the alliteration makes the story sound like a soothing melody.
"There was the sound like a great matted yellow hive of bees within a dark bellows, the lazy bumble of a purring lion."
At other times the alliteration can intentionally make the audience feel exhausted.
"...it had shut up its windows and drawn shades in an old maidenly preoccupation with self-protection which bordered on a mechanical paranoia."
It can make the reader feel the same sense of urgency and efficiency that the house feels.
"If a sparrow brushed a window, the shade snapped up. The bird, startled, flew off!"
And near the very end of the story, as the house is desperately trying to save itself, the alliteration magnifies the sensation of doom and inevitability that surrounds the house's valiant efforts.
"While scurrying water rats squeaked from the walls, pistoled their water, and ran for more. And the wall sprays let down showers of mechanical rain. But too late. Somewhere, sighing, a pump shrugged to a stop."
In this story, the alliteration teaches the reader what to feel, what to think, and how to predict the horrible fate that this house faces.