I think the full message of "There Will Come Soft Rains" is not fully appreciated without reading it in its context within The Martian Chronicles, but it does make several salient points in a stand-alone format.
In the broader context of TMC, humanity has been making difficult but progressively more successful expeditions to colonize Mars, discussing its own fate and the fate of the Martians in the process, but failing to account for the corrupting effects of the same society that made the colonizing technology possible in the first place. Where the Martian expeditions are in some ways, an exploration of a frontier and the very limits of human achievement and philosophy, "back home" things are the same as they ever were, and a nuclear war is foreshadowed repeatedly.
"There Were Come Soft Rains" depicts the aftermath of this inevitable war; the entire edifice of human civilization collapses in a matter of days or months. Many of the Martian colonists return to Earth out of emotional sympathy, only to presumably die with the rest. By the end of the anthology, there seem to be very few humans left alive on either planet.
The message of "There Will Come Soft Rains", either by itself or fitting into this overall narrative, is that humanity was so concerned with its own importance, its conveniences, and its comforts, that its own demise came as a surprise, but that nature doesn't really care, and our sense of importance was relevant only to ourselves. Gradually, nature begins to reclaim the Earth from us and our ruins, and our existence sudden seems to be very superficial.