The significance of the spots is that they represent where the members of the family died.
The image is significant because it describes the people who used to live in the house, memorialized permanently by the atomic bomb. When the bomb exploded, the man was mowing the lawn, the woman was picking flowers, and the children were playing with a ball. The bomb blast turned them each into a “silhouette in paint.”
The five spots of paint-the man, the woman, the children, the ball-remained. The rest was a thin charcoal layer.
Until this part of the story, all we know is that it is the year 2026 and the house seems to be automated, but the people it is supposed to take care of no longer exist. When the dog arrives in such bad shape and dies, only to be incinerated like garbage by the mice.
The irony of the story is that technology is so advanced that it makes human life easier, because the house keeps itself clean and meets every whim of its inhabitants, but it also took away human life. The atomic bomb is also technology, and it killed all of the humans.