It's already explained in the other areas. What are you confused about?
basically the family is just a small family, where the scientist that worked with the people who shut the power off is either a parents or a very close (But olderish) relative.
The kids would be just old enough to hardly remember the black out, and wouldn't know the scientist is a scientist.
In the show, the mother was the scientist, and the day before the power went out, she went on the run and stole the technology they had made because the company she worked for was planning some very evil stuff with the devices.