Black needs four moves to promote the black f-pawn, if white doesn't stop it with his king. White needs also four moves to promote. I.e.,
[Variant "From Position"]
[FEN "8/7p/4k3/1p2ppP1/1P5P/8/P1K5/8 w - - 0 1"]
1. h5 f4 2. g6 hxg6 3. h6 f3 4. h7 f2 5. h8=Q
But, black stands in its square and in reality this would happen:
[Variant "From Position"]
[FEN "8/7p/4k3/1p2ppP1/1P5P/8/P1K5/8 w - - 0 1"]
1. h5 f4 2. g6 hxg6 3. h6 Kf7 4. Kd3 Kg8 5. Ke4 Kh7 6. Kf3 Kxh6
where black simply picks up white's free pawn and then goes on to assist his three connected free pawns.
Fortunately, white also has a pawn majority on the Queen's side, which also costs four moves to promote:
[Variant "From Position"]
[FEN "8/7p/4k3/1p2ppP1/1P5P/8/P1K5/8 w - - 0 1"]
1. a4 bxa4 2. b5 f4 3. b6 f3 4. b7 f2 5. b8=Q
But black also stands in that square. Black can not stop both, the b-pawn and the h-pawn however:
[Variant "From Position"]
[FEN "8/7p/4k3/1p2ppP1/1P5P/8/P1K5/8 w - - 0 1"]
1. a4 bxa4 2. b5 f4 3. b6 Kd6 4. h5 f3 5. g6 hxg6 6. hxg6 f2 7. g7 f1=Q 8. g8=Q
That is the best computer line and now it is a draw due to infinite check.
Note that black must play 1... bxa4 or white simply promotes the a-pawn, stopping blacks pawns with his king: also black needs the threat of two pawns promoting.
On top of that, white must play 2... hxg6 and not h6 as before. For example (the best computer line):
[Variant "From Position"]
[FEN "8/7p/4k3/1p2ppP1/1P5P/8/P1K5/8 w - - 0 1"]
1. a4 bxa4 2. b5 f4 3. b6 Kd6 4. h5 f3 5. g6 hxg6 6. h6 f2 7. h7 f1=Q 8. h8=Q Qf2+ 9. Kd1 Qd4+ 10. Ke1 Kc6 11. Qb8 Qxb6 12. Qe8+ Kd5 13. Ke2 Qc6 14. Qd8+ Kc5 15. Kf2 Kc4 16. Qd1 Qc5+ 17. Ke2 a3 18. Kf1 e4 19. Qc1+ Kb5 20. Qa1 Qc4+ 21. Kg1 a2 22. Kh1 e3 23. Kg2 Ka4 24. Qd1+ Ka3 25. Qd6+ Kb2 26. Qb8+ Kc2 27. Qa8 e2 28. Qa5 Qe4+ 29. Kf2 e1=Q+ 30. Qxe1 Qxe1+ 31. Kxe1 a1=Q+ 32. Kf2 Kd3 33. Kg3 Qd4 34. Kh3 Qg1 35. Kh4 Ke3 36. Kh3 Kf3 37. Kh4 Qg3#
where white avoided queen trade as long as possible (pointlessly). The actual problem being that if white would win one pawn, then black would be able to immediately trade queens and easily win with king plus two pawns against king. For example:
[Variant "From Position"]
[FEN "8/7p/4k3/1p2ppP1/1P5P/8/P1K5/8 w - - 0 1"]
1. a4 bxa4 2. b5 f4 3. b6 Kd6 4. h5 f3 5. g6 hxg6 6. h6 f2 7. h7 f1=Q 8. h8=Q Qf2+ 9. Kc1 Qc5+ 10. Kd1 Kc6 11. Qa8+ Kxb6 12. Qxa4 Qd4+ 13. Qxd4+ exd4 14. Ke2 Kc5
or any of the other billion possibilities. It is however crucial to understand what K+Q+3 pawns against K+Q has infinitely larger chance to win in this manner than K+Q+2 pawns against K+Q.
Also keeping it a draw with infinite check is an art by itself that isn't very trivial. In general you just want to keep the queen on the board and keep giving check. Try to play this against a computer to see all the pitfalls.