The house stood alone on a street where all the other houses were rubble and ashes.
In addition to the above answer, I might just add that the line also actually introduces an important character in the story: the house. The house has survived the holocaust even though the inhabitants are nothing more than five spots covered by charcoal on the wall. The house carries on as though the horrors of human madness can't touch it; as though it were an individual, really--the last individual.
The return of the dog is another thread of the idea of the last survivor. Yet, the dog is dead within an hour. The house too, the last survivor in a decimated neighborhood, will die within its own metaphorical hour, which comes soon with the upwelling of the wind of nature. This ties back to the dog which is by definition a part of Nature. The summation is that neither nature nor machine can withstand and survive the genius of man turned to madness.
In my opinion, the main importance of this line is that it sets the scene for us. In other words, it tells us what has happened to the city where the house is and, more importantly, to the whole society.
One of the major points that Bradbury is trying to make in this story is that people are overdoing it with their technology. He shows this by juxtaposing this wondrous house that does everything for you with the city that has been levelled by other kinds of technology.
So, by writing this line, Bradbury is showing us that the city has been destroyed. It is important for us to know this because that brings home a major theme of the story.