Here's what I try to pick up from GM games:
1) Key maneuvers and plans in a particular setup. This could be as simple as in a certain line it's best to play Kh1 before pushing your f-pawn to avoid tactics with a check along the g1-a7 diagonal. It's best to try to understand why the maneuver was made and why it makes sense in this position, and also trying to decide why it may not be necessary in an alternative position. Often, annotations can fill in some of the gaps here, but try to get a firm understanding of why.
2) Evaluating which side is better. This is one of the best abilities to have in chess. As you're going through the game, try to decide which side is better and determine what advantages makes his position better. In some of these GM games, it can be as small as a weak pawn on c6. If black is allowed to push it to c5 and trade it off, then he has equalized and the advantage that gave white the better position has evaporated. Once you understand these advantages, notice how the GMs will play to preserve or even enhance their advantages, while their opponent will try to eliminate theirs. I just looked at a Kasparov game in the Catalan, where white had a strong setup with pawns on d4, e3, f2, and f4 which created a central bind. Black's plan was to break up white's pawn center with either ...c5 or ...e5, so White's moves centered around preventing those pawn breaks or insuring that these moves would lead to other problems in the position (discoordination of black's pieces once things open up).
3) Picking up on buzz words and understanding those concepts. Notice when the annotations mention for example flexibility (playing a necessary move first, and leaving your options open for other pieces) and initiative (control/grip on the position).
Another nice concept you'll notice is avoiding complications. This involves playing for the simplest win when you have the better position, and declining offers by your opponent to complicate the game. By complicating the game, I mean adding additional advantages and disadvantages to each side. The simplest game is when you have a distinct advantage. To complicate it, black may give up some material to gain other advantages that require you to play more accurately.
4) Don't overly trust the analysis. Make sure you prove it yourself. If some string of analysis mentions that white has an advantage, set up the position and make sure you understand why white has the advantage there. Maybe even play a few moves against yourself (or a computer perhaps) to see how the game might develop from there. An important part of chess is gaining the advantage, but equally important is converting that advantage into a win.
5) Try to decide if you would play the same moves in your games, and understand whether the GMs move makes sense for that position. Of course you can't blindly just play similar moves if the position is different in your games, but get a feel for what type of moves a position might demand. For example, if you have a weak pawn, improve it or trade it. If you have the bishop pair open the position. If you have knights try to keep the position closed or find an outpost.