Sara Teasdale's poem was an inspiration for Bradbury's story of the same name. In part, he includes the poem as tribute to Teasdale, but more importantly, because it underlines his main theme or message: that humankind should exercise more humility and care in its use of technology. In the story, a programmed house goes on functioning after a nuclear attack has killed everyone living in it. The house is not capable of caring whether the humans it is serving are alive or dead.
Teasdale makes a similar point in her poem, when she says that nature would not mind or notice if we humans were extinct. We humans should remember we are completely expendable as a species and not all that important in the nature's grand scheme. Bradbury is trying to make that point crystal clear by including the poem, the last lines of which read:
Not one would mind, neither bird nor tree
If mankind perished utterly;
And Spring herself, when she woke at dawn,
Would scarcely know that we were gone.
One of the ironies of the story is the house reads the poem to Mrs. McClellan after she has perished in a nuclear holocaust. The house notes—to nobody—that Sara Teasdale is one of Mrs. McClellan's favorite poets. What Bradbury is saying is that we would do well to really listen to what poets like Teasdale—or writers like himself—are trying to communicate and reduce our reliance on technology before it is too late.