The dog is dying because it is one of the last creatures left alive after surviving a nuclear blast.
The dog is in very bad shape. The world has undergone a nuclear event of some sort, so that the inhabitants of the house were reduced to spots.
The five spots of paint—the man, the woman, the children, the ball—remained. The rest was a thin charcoaled layer.
That is all that remains of the family that used to live in the house. The spots are where they were standing when the blast hit. The dog survived though, but he is suffering the effects of nuclear exposure. He is dying. Sadly, he tries to get into his house and find his family, but he doesn’t realize they are dead.
The dog ran upstairs, hysterically yelping to each door, at last realizing, as the house realized, that only silence was here.
As the house makes pancakes for a family that no longer exists, the dog dies. The house’s electronic mice come and clean it up.
This is significant because there are no human beings left, and there are now no living creatures left. The robots have taken over. They have no sympathy when the dog dies, sweeping it up like garbage even though it is a living creature. It is remarkable that the dog survived so long, but it gets no credit for its achievement. Its only reward is the sad realizable that it has outlived the family it loved, and is trapped in the automated emotionless world of the house.
Like many of Bradbury's stories, this one tells a cautionary tale of technology versus humans. The most touching moment in this story, or perhaps the most jarring, is when the dog's body is removed by the robot mice who do not distinguish it from other garbage. The contrast between the dog crying for his missing family and his realization that they are not coming back and the mice, who used to make the family's life easier, throwing him away, is a sharp one. The very technology that makes human life easier can also destroy us. It is what made the bomb that killed everyone but the robots, we assume.