I remember the first time reading this story. I thought it was so cool. I didn't quite understand the poem, and I definitely didn't understand the poem as it compared with the story. Then my English teacher started in on both, and I realized that the poem and the story have a lot in common. The other thing that I noticed is that both items have bleak and depressing messages.
The poem ends with the following four lines.
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.
Basically, Teasedale is saying that nature itself could care less whether or not mankind walks the surface of Earth. Nature will carry on its business. It doesn't need humans and will barely even notice our complete absence. That's sad. When a reader compares that message to the full story, then a reader realizes that Bradbury's message is the same. Mankind (at least in Allendale) has been wiped from the face of Earth. The house goes about its business as if people were in it. The house does ask the missing people questions, but the house eventually just makes its own decisions. Ultimately, it doesn't need the people to operate. As the story ends, the house is dying. Nature is taking it back. Nature will be taking back the entire city. Given enough time, the processes of ecological succession will ensure that all traces of human habitation are erased. Both the story and the poem point out the fact that the world is perfectly capable of going about its business without humanity in it.