It was the night of my Day 13 survival. I was sitting at my campfire which I just fed with a log (so it was burning brightly), I had just ate to full, and I had nearly full sanity and health. All of a sudden, this giant shadow hand appears and starts grabbing towards my campsite. Without warning my fire goes out and I die. If I had full stats and a brightly lit fire, why did the hand appear? And how was it able to kill me?
The giant hands are called Night Hands. They also have a kind of creepy music box like music that plays to signal their appearance. What they try to do is put out your fire. What I imagine happened is that it successfully put out your fire and you were then eaten by a Grue. The chance for them to start spawning is when your Sanity level is below 82.5%, and, like all Shadow Creatures, they'll occur more frequently the more insane you've gone.
You can try to drive them away by running around and stepping on them. Keep in mind that this will cause your sanity to go down, so you'll have to keep an eye on it. You'll know that the hand has completely disappeared if you don't hear the music anymore. One thing that helps a bit is running out into the darkness a little to make sure that you've chased them as far as you can, to the point you're pretty much in complete darkness. I've found that this helps get rid of them sooner.
As a safety measure, you can try to stoke your fire to the max in order to make it so it's more difficult to put out, but this will also burn a lot of resources and doesn't actually make them go away.