This position is example 7.57 on page 285 - chapter ROOK VS MINOR PIECE(S): R. Dautov - T. Heinemann German Ch. (Bremen) 1998. I was able to find that game online on Chess Tempo and on Lichess:
[Event "70th ch-GER"]
[Site "Bremen GER"]
[White "Dautov, Rustem"]
[Black "Heinemann, Thies"]
[FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"]
1.d4 Nf6 2.c4 c5 3.d5 b5 4.Nf3 bxc4 5.Nc3 d6 6.e4 g6 7.Bxc4 Bg7 8.e5 dxe5 9.Nxe5 O-O 10.O-O Ne8 11.Re1 Nd6 12.b3 Qc7 13.Bf4 Nd7 14.Nxd7 Qxd7 15.Be5 Rb8 16.Qe2 Bxe5 17.Qxe5 Rb4 18.Ne4 Nxc4 19.bxc4 Rxc4 20.d6 Rxe4 21.Qxe4 exd6 22.Rad1 Bb7 23.Qe7 Qc6 24.f3 d5 25.Rb1 Bc8 26.Qxa7 c4 27.Qd4 Be6 28.Rb6 Qc7 29.a4 Rc8 30.a5 Qa7 31.Ra1 c3 32.Rb4 Qa6 33.Qb6 Qd3 34.Qd4 Qa6 35.Rb3 Qc4 36.Qxc4 Rxc4 37.a6 c2 38.a7 c1=Q+ 39.Rxc1 Rxc1+ 40.Kf2 Ra1 41.Rb8+ Kg7 42.a8=Q Rxa8 43.Rxa8 h5 44.Ke3 Kf6 45.Kd4 Kf5 46.g3 Kf6 47.Ra3 Kf5 48.Re3 Kf6 49.Re5 Ke7 50.Kc5 Kd7 51.Re3 Ke7 52.Rd3 Kf6 53.Kd6 Kf5 54.Rd4 g5 55.h4 Kf6 56.hxg5+ Kxg5 57.Ke5 Kg6 58.Rb4 Kg7 59.Rb8 d4 60.Kxd4 Ba2 61.Ke5 Bc4 62.Kf5 Bd5 63.f4 Bf3 64.Rb6 Bg4+ 65.Kg5 Bf3 66.f5 Bg4 67.f6+ Kh7 68.Rb7 Be6 69.Kxh5 Bd5 70.Rb4 1-0
The example, according to the book, demonstrates the difficulties a bishop faces in general (even with an extra pawn!) against a rook. After 43... h5!? 44. Ke3 Kf6 45. Kd4 Kf5?! 46. g3 Nf6, the book analyzes some variations that could occur if 46... h4!? was played.
55. h4! is the finishing touch: it creates more weaknesses on the dark squares. Black has lost.
While it might be feasible for an engine to draw (playing black), it's almost impossible for a human to make no mistakes, as these positions are too tricky. The more the pawns on both sides the greater the superiority of the rook and the better the chances to win.