What do chess champions see that we do not?

http://www.chessgames.com/perl/chessgame?gid=1102375 good example of what Chess master Aron Nimzowitsch was thinking while playing this game. 

What stands out more to me in all these posts is that memory appears to be a major factor or component in playing better chess. When I read that they draw from experience, this is memory, decision-making and understanding is it not memory again?

It's their mother tongue again memory, no?

Strong players see a lot of patterns on the chessboard that triggers ideas stored in their long-term memory.

They see the memories of all of the thousands of games that they have played that you have not, and also the dozens of chess books they have studied.

It would appear to me that memory retention and recall is a very important factor in playing chess whether consciously or subconsciously.
How does one develop this attribute?

And the biggest question is with all of those games memorized and all of the positions played over thousands of games played how does one extract the exact favorable combination from one single game or 2 to see the route and moves to play?

You don't 'extract' relevant patterns, as much as they 'jump in your face' so to speak. For masters, some moves are as evident as 9*9 = 81. For less experienced players, they wouldn't notice the '9' in the position in the first place.

Still, titled players still have to find a way to apply known patterns in a specific position : here they mainly use 'calculation' (if I do this, he does that, then I do this...), and various heuristics (logical thinking, selecting the best moves by eliminating all the other plausible ones, etc.)

seconded

They have intuition