I think everyone is correctly describing what happens, we just have different ways of describing it. This is how I explain it.
You learn to anticipate better. You start to react when a hint of motion is present. You react to sounds much better. You must learn things that you might not learn in a car. You see a hint of motion inside a parked car that may be a door about to open.
You hear the door latch you could not hear if you were in a car. You do have better visibility. You can hear cars coming up from behind, even if you need to concentrate on a problem in front of you. You can hear cars on side streets sometimes, before you see them. A lot of this gets programmed into the subconscious. You don't always know it's working, but you can't even shut it off. It's a sub routine going in the background like a firewall protecting you, while you pay attention to the obvious things at the same time.
The perfect example is slvoid's description of responding to a skid on the ice and recovering. You can not think that through, by that time you would be on the ground. The process is learned, stored and switched to automatic and it happens so fast you don't even know how you did it. After it happens you think about it, then you may get scared. At that point it does not matter.
It's similar to the hunter recognizing that an animal was here because of the grass being flattened where he lay down. A lot of people would just miss that. Or the hunter realizes that all of sudden all the small birds and animals are quiet, either they noticed him, or another predator is near by. All the birds take flight when the hunter has been still for an hour, something else is coming. It's like the Indian scout feels the vibration in the ground as the stampede of buffalo is a long way off. No one else understands.
A cyclist picks up the same things. You see a car or pedestrian react oddly to something you can't even see, like a car on a side street, you get ready to slow or stop.
I believe the term correct technical term for this is "Situational Awareness". It does vary by individual as well.
I realized one day riding in traffic that I have an invisible safety space all around me in all directions, when something gets into or close to that space I just feel it. In front it's my stopping distance, on the sides it's an awareness of closeness if someone were to move sideways. In the back in my mirror it's if something is too close for me to slam on the brakes. I rode a long way with a friend in heavy traffic the other day and he wanted to be close enough to talk in heavy traffic. It messed me up all day and was very annoying.