Patterns. From thousands of master games. A level of pattern recognition that gives him a sense about what to do in a position. This is because he has been doing it since he was a child.
what does magnus carlsen see when he sees a chessboard

Yeah, it's pattern recognition. It's like looking at a page of written text.
Illiterate - non players
Can recognize which language it is - knowing how the pieces move
Recognize some of the characters - knowing some basics like develop
Recognize elementary words - Social players and weak club players
Can read books for young children - average club players
Can read a newspaper or magazine - strong club player
Can read technical papers and interpret literature - weaker titled players
Can write technical papers and literature - the best players.
Writes some of the best literature of the day - Carlsen.
Something like that.
Back in the early 90's I played in a chess tournament in Chattanooga and one round I faced NM Jerry Wheeler from Nashville I think. I was only rated class D or low Class C and it was obvious that any chance I had of winning a prize went up in smoke after realizing he was my next opponent.
I remember him telling me that I should look at that game as a learning experience because many chess players will never have the opportunity of playing a real master over the board. He then told me after he handed me my hat that he couldn't explain it but the more chess games you play you begin to recognize chess patterns and that would come with experience. As weak as my pattern recognition is I now understand what he was telling me.

I think this question really can't be answered accurately unless you happen to be Magnus Carlsen, or perhaps a psychic.

Back in the early 90's I played in a chess tournament in Chattanooga and one round I faced NM Jerry Wheeler from Nashville I think. I was only rated class D or low Class C and it was obvious that any chance I had of winning a prize went up in smoke after realizing he was my next opponent.
I remember him telling me that I should look at that game as a learning experience because many chess players will never have the opportunity of playing a real master over the board. He then told me after he handed me my hat that he couldn't explain it but the more chess games you play you begin to recognize chess patterns and that would come with experience. As weak as my pattern recognition is I now understand what he was telling me.
Are you still a class C player after 20 years?
I think this question really can't be answered accurately unless you happen to be Magnus Carlsen, or perhaps a psychic.
I believe all strong chess players recognise chess patterns. I read one time that the reason Siegbert Tarrasch never became World Champion is because he was weak at math.. I believe it may also explain why once boys reach puberty they start passing girls when it comes to chess. Men and women do think differently.
Back in the early 90's I played in a chess tournament in Chattanooga and one round I faced NM Jerry Wheeler from Nashville I think. I was only rated class D or low Class C and it was obvious that any chance I had of winning a prize went up in smoke after realizing he was my next opponent.
I remember him telling me that I should look at that game as a learning experience because many chess players will never have the opportunity of playing a real master over the board. He then told me after he handed me my hat that he couldn't explain it but the more chess games you play you begin to recognize chess patterns and that would come with experience. As weak as my pattern recognition is I now understand what he was telling me.
Are you still a class C player after 20 years?
My 4 year old son was killed in a car wreck and I quit playing chess. I played in my first chess tournament in 18 years a couple of months ago.

I believe all strong chess players recognise chess patterns. I read one time that the reason Siegbert Tarrasch never became World Champion is because he was weak at math.. I believe it may also explain why once boys reach puberty they start passing girls when it comes to chess. Men and women do think differently.
I think you didn't pick up on the joke. While what you say is probably correct - men and women do, as a rule, think differently (no sexism intended, that's simply how the reality is), and Tarrasch having poor "calculating" abilities could hinder his strength, my post wasn't intended as a serious reply (several of which had already been posted).

Magnus is a walking chess library. He can recall thousands upon thousands of master games, recognise individual positions from them, name the players, their history, name the year and tournament the game was played at etc.
When you throw an idea at him he instantly has a good idea of what it's about and what's going to result from it.
Most of us aren't so good at that.
And most of us are ofcourse outright unwilling to put in that kind of effort. (Ie. a life time all consuming study of chess.)
Plus ofcourse he is extremely good (ie. perhaps best there ever was) at calculation and every other skill required for chess..

what do his eyes/brain see that the rest of us don't
A whole lot more than i see
He sees a chessboard with pieces on it and visualises positions in his head, I presume.
Off topic: the title of this thread reminded me of the following picture:
http://3.bp.blogspot.com/-epzbWI-GsnI/UceOjyBOMgI/AAAAAAAABK0/7Taijj6dOMo/s1600/chess+meme+how+we+see+ivanchuk+ceiling.jpg
what do his eyes/brain see that the rest of us don't