Airports never close. Pilots can choose to land or not based on the reported weather. Individual runways close to have the snow plowed off, but the reason for the closure isn’t that the snow depth exceeds some standard, but rather that the runway becomes covered with snowplows and hitting one would be bad.
Generally speaking, the worst acceptable weather allows the reported horizontal visibility to be as low as 300 feet or 100 meters. Which at touchdown speed is seeing about 1 second into your future. There are no ceiling height limits associated with such approaches / landings, so the fog can reach right down to ground level. In a big airplane your face is 40-50 feet off the runway when the main gear touches down.
These approaches and landings are flown by the autopilot with the pilots monitoring. If anything on the instruments or out the window looks crappy you go around. A very late decision to go around will result in the airplane bouncing off the ground. If there was an empty runway under you, that’s not a cause for alarm. If instead it was a car park or a freeway or an occupied runway that would be problematic.
The “whoop whoop” at the end is the autopilot disconnect alarm. As you get down to taxi speed you disconnect the autopilot and begin steering manually. The noise ensures everyone in the cockpit is aware of the transition. It also sounds if the autopilot disconnects on its own due to malfunction. You can imagine the Keystone Kops event that could happen if both pilots thought HAL was flying and HAL thought the pilots were flying.
At the end we can see visibility is about 300 feet. The white lights at runway edge are 100 feet apart. If you can see one directly alongside you, plus the next 3, you have 300 feet. On the film we see about 1-1/2 lights ahead, but the brightness is such that we can’t pick out any more. I bet, based on experience, that they could see another 1 or so.
It’s also commonly the case that the near end of the runway has better visibility as the fog is stirred up by all the aircraft, and warmed by all the exhaust. At the far end, or during taxi, it’s not uncommon for visibility to get worse.
Getting lost while taxiing is another way to become famous. Many people have been killed over the years in accidents caused by a lost airplane taxing onto an active runway.
“Awesome” and “nailed it” are comments by whoever posted the video, not by the guys doing it. For them it’s just another day at work. Not that many airports get dense fog. But the places that do (e.g. London, Frankfurt, Seattle, San Francisco) tend to get it often. So the crews that frequently transit those places get real good at their role in low weather ops.