Bradbury doesn't tell us directly what happens to the family. However, we're given vivid indications via images.
Look at what happens as the house works through its day. At 10:15, we're told that the side of the house is black, except for the five silhouettes of the family. These images, we're told, are "burned into the wood."
Now go back to 10:00. The house is the only one standing in "a city of rubble and ash." We're also told that the city glows at night, due to radioactivity.
We have to conclude that the family--and likely everyone else in the city--have been killed by a nuclear war. They were burned to death, incinerated, even vaporized, by a flash so intense that it burned their images onto the sides of their house.
That's the sadness of the story: the house goes on after its people died.