White get compensation in the centre (has two central pawns against Black's one) and tempo (Black needs to use time to keep the pawn - if possible at all). It is a common theme in the Cathalan Opening e.g.
I you (like me) are a pawn grabber who love to win a pawn up endgame it is probably not openings for you unless you play them to learn.
I you (like me) are a pawn grabber who love to win a pawn up <snip>...
Actually I gravitate towards the opposite: I'd rather give a pawn for compensation, but it's difficult for me to find the compensation. Especially (see original diagram) the side variation where white responds with a4, b3 or Nh4. My point is, how does white exploit the compensation (which is 2 pawns in the center like you mentioned). I'm talking helicopter view here.
Such as "White has compensation due to his two center pawns vs one and therefore white can/should ..."
A strong center is not something that you have to exploit quickly. It will be with you for the most/whole game and give you an easier game e.g. easier to move pieces from one side of the board to the other. It is not like a gambit were you have to exploit your tempo.
So you should basically just play like you are not a pawn down. Just play on the whole board.
Whether black plays 4...e6 or 4...Nf6, 5.c4 is a fine move. But I can't wrap my head around why this works after 5...dxc4.
In one variation as you can see, 6.Na3 is the proper response. In the other variation multiple approaches are entirely feasible. But none of them will get the pawn back (fast). In the Na3 case, the pawn is won back, but I don't understand how this advantageous for white. What thing in the opening play made this approach OK?
In many openings I'm constantly struggling to understand this. Can somebody please educate me how to think here. What to look for, and so forth. Really, I'm so much struggling with this typical concept of playing c2-c4 like this.