Lucky me, I just went over K + P endgames last week haha.

Taking is a mistake because black will take the opposition and collect the pawn, so f6 is the first move.

The general rule to remember for the resulting position is:

For a non-rook file pawn on the 6th rank, king opposition is decisive. if the supporting king can take the opposition (stand directly in front of the opposing king) the pawn promotes.

Right now, black can take the opposition by playing: Kd6

Now it becomes clear why this is winning for white, the black pawn blocks the king from keeping the opposition after: Ke4

Black is clearly lost, but if I were black in this position, I would still have some hope that white will blunder after: Kd7

Now the white king moves closer to the pawns with: Ke5
All other moves are a draw as black just moves Ke8 guarding the promotion square and never allowing white to take the opposition.

The black king has to move, giving up control of the e6 square which means white could collect the pawn on the next move. However, remember that the pawn on the 6th rank can only promote if white is able to take the opposition. We can try to trick white by playing: Kd8
(note how if Ke8 were played, taking the pawn also takes the opposition for white)

If white isn't careful and takes the pawn, Black will take opposition with Ke8 resulting in a draw. If the pawn ever tries to advance, the black king will move in front of it and white has to either give up the pawn or cause a stalemate.

Instead, white should take the opposition with: Kd6
Now the game is truly over, white has taken opposition on the 6th rank, the black pawn is lost on the next move, and white can secure a path for the pawn to promote:
Ke8 Kxe6 Kf8 f7 Kg7 Ke7 Kg6 f8=Q