Pawn endgame advice please...

I'm working on pawn endgames at the moment, yet another noob desperately trying to improve, and came across this one. 

White to move, Kd2 is the move. 

Now, the puzzle states that white wants to reach the 3rd row and black the 5th, but both 'want to achieve these goals after the other guy'.  

Is it possible someone can explain to me just why it has to be 'after the other guy' please? Is there something obvious here, transferable to similar positions, that I should be making sure I grasp?  

I'm putting in the effort to get better and I'm trying to glean as much advice from folks as I possibly can, so thank you in advance for any pointers.

*never posted a game before, so if it hasn't worked  positions are Black K on d6 and pawn on f6, White K on c2 (white to play). 

I'd suggest you read some basic K+p vs K endgame theory. If you get a good grasp on the basics, the answer will become obvious to you.

IMBacon has explained it pretty well. King and pawn endings are very delicate, and having the better king is often decisive. When to kings face each other, the player who doesn't have to to move is said to "have the opposition," because the player who has to move must give ground with his king and let the other king advance. In the following examples, if it's white's move, he loses, if it's black's move, it's a draw!

White to move

Black to move

 Clear explanation, thank you.

Thanks for the advice folks, much appreciated. 

Glad to help, and good luck!