Classic, drawn game, no matter who has the move. White's only hope is to secure the queening tile by landing his king in the vicinity (e.g. f7, f8, d7 or d8). This, however, is entirely preventable.
This (and Kd4) is the only way to progress. Failed tries: 1.Kf3,Ke5 2.Ke3,Ke6 (and we're back to where we started); 1.e5??,Kxe5 draw
Black takes the Opposition and prevents further advance of white's king.
The logical continuation. Failed tries: 2.Ke3,Ke6 (and we're back where we started)
Surrenders Opposition but still prevents white's pawn and king from advancing.
The only move; anything else and the game is drawn after ...Kxe5.
! The critical move and the end of the cycle! Notice that this is the same position as we started but one rank closer to the endzone. The sequence of 5 half-moves leading to this point can be repeated over and over (should the white king have started on e1 with pawn on e2) with the same drawn result. Both ...Kf7 and ...Kd7 are FAILS for black: ...Kd7?? 4.Kf5,Ke7 5.e6,Kf8 6.Kf6,Ke8 7.Ke6 and now it's obvious white can secure the endzone (a)...Kd8 8.Kf7 1/0 (b)...Kd7 8.e6+,Ke8 9.e7,Kd7 10.Kf7 1/0