The key to the win: Move the King, not the Pawn!
The King must clear the way for the Pawn to come through. He does this by taking the opposition: he moves to a face to face position with Black's King, and forces Black to give way. White can then seize control of e8, the square on which the Pawn promotes to a Queen.
Without this preparation, advancing the Pawn does not work. For instance: 1. e6, Kd8 2. e7+, Ke8 3. Ke6 (to protect the Pawn) and Black draws by stalemate.
The proper technique:
1. Ke6!
The Kings face each other, and White having the opposition, forces Black to give way. The King that can compel the other to yield ground either by retreating or stepping aside, is said to have the opposition.
This is now for White the ideal position in King and Pawn endings. He wins no matter whose move it is.
- ..., Kd8
- Kf7
From this square, the King not only controls e8 (the square on which the Pawn becomes a Queen) but is also in position to escort the Pawn on its way up.
- ..., Kd7
- e6+ Kd8
- e7+ Kd7
- e8=Q+
White wins.