In the starting position you give it is easy for white to create a passed pawn on the kingside and more difficult for black to do the same on the queenside. To guarantee that black cannot create a queenside passed pawn you need to aim for this position:
[fen "4k3/1pp2ppp/p1p5/8/P1P1P3/1P6/5PPP/4K3 w - - 0 1"]
From here the win is still not easy or guaranteed but black has no prospects whatsoever and just has to try and find a way to draw.
White can ignore all black queenside pawn moves apart from captures. If black captures on the queenside then white recaptures leaving, again, a structure which does not allow the extra pawn to generate a passer. Otherwise white doesn't touch the queenside pawns and can concentrate on creating a kingside passed pawn.
Once a passed pawn is created white tries to push it as far up the board as possible before abandoning it to capture the queenside pawns while black captures the passer. Black's job is to stop this plan.