While our sun is a yellow dwarf that's host to 4 terrestrial planets and 4 gas-giants, it's far from a "typical" star system. In fact, our solar system is pretty rare in the fact that it doesn't reside in a binary (two star) system; the vast majority of stars in the universe actually reside in multiple-star systems. Besides that, we've seen plenty of strange solar systems that have hot-Jupiters orbiting their parent stars over 1/10th as close to their star as Mercury is to the sun. We've seen a terrestrial planet that orbits its parent star with a stellar companion: a gas-giant that's as close to that star as 5 times the distance between the Earth and the moon. Basically, what we've learned is…we need to expect anything and everything. The universe tends to not give a crap about what we perceive as "impossible." Such as this: a star system that has not one, but two stars, that are locked in a tight orbit from a close distance of only 600,000 km (360,000 miles)! They are so close that one of the stars eclipses its counter-part every 3 hours and seven minutes.
This system leaves us with far more questions than answers. First, why didn't the white-dwarf engulf its counter-part during the red giant expansion stage? Second, how did the PLANETS survive the expansion? As we've seen in the past, stars in the red giant phase of stellar evolution lose tons of mass from all of the matter being expelled into space. This causes its gravity to get weaker, thus its hold on its orbiting stars would weaken, causing them to migrate outwards- perhaps they would even be ejected from their system entirely. There is a theory though: the two planets were formed AFTER the red-giant expelled most of its material out into space!
As for what you would see if you were living on one of these huge planets or one of their moons, I think the bad astronomer himself (Phil Plait) says it far better than I could:
"From the inner of the two planets, the two stars would be a bit less than a tenth of a degree apart; about 1/5th the width of the full Moon. You’d be able to see them as separate stars. The red dwarf would barely resolve itself as a disk; it wouldn’t look like just a dot in the sky. The star is far less luminous than the Sun, but would still shine about 20 times brighter than the full Moon on Earth. In other words, if it were the only object in the sky you could read by it, and looking at it would make you squint a bit.
The white dwarf, on the other hand, is tiny: only about 30,000 km (roughly 20,000 miles) across. It would be a dot in the sky from that distance. However, it’s so hot that it shines more brightly than the Sun does, and from that inner planet would be about half as bright as the Sun appears to us from the Earth. It would be an intense pinprick in the sky, a brilliant dot that would be very painful to look at. In fact, it would drown out the red dwarf completely, shining thousands of times more brightly. "