1. It is very important to grab as much space as possible in the ending, if you can. Therefore, ideally, white would love to set up his pawns on g4 and h4 right now, severely limiting black's movement on the kingside.
2. If he starts with g4, black can just respond with g5 and the expansion on the kingside is stopped, because after f4 black can just hold his ground with something like Nh7 or even take on f4 and get rid of his doubled pawns.
3. Therefore, he starts with h4, preventing g5. Nh5 is not scary, so g4 is incoming next move, slowly building the pressure and limiting black's options.
This isn't an instructive position, but a beautiful game. This article is from chess.com and showcases the game Timman-Ernst from Wijk aan Zee's B group this year. This is the type of game I could only dream to play.
That looks like a dead draw to me, I don't see how white can make progress. Since black's king can't be dislodged from c8, the white king can't ever participate in the game. All black has to do is not lose the g5 pawn and white can't do anything.