Probably forgot (or didn't know) the "en passant" rule. It's very difficult for either side to make progress here.

Preferable might have been 37.Re1+, with the following forcing line in mind: 37...Kf7 38.Rxd8 Bxd8 39.Rxe7+ Kxe7 40.b6 and if 40...Kd7 41.b7 Kc7 42.Bb6!! Naturally, if 41.Bc7 instead, then 42.Ba7. Suppose, then, Black recaptures on e7 with the bishop? 39...Bxe7 40.b6?! Bd6 41.b7 Bb8, with a difficult defence. But White has something better: 39...Bxe7 40.Ba7!! Note that this won't work after the King recapture as 39...Kxe7 40.Bxa7 Kd7 41.b6?? Kc6 stops the pawn cold. Finally, a glance at the intermezzo after White takes on d8: 38...Rxe1+ 39.Kxe1 Bxd8 40.Ba7 as before. The Black King is too far off to contain the passed pawn. But it has to be admitted: the text move is almost as attractive!

This is a very late reply to your comment, but I'm afraid the bishop guards e1.

How ineffectual is 44.Rc6? If [A] 44...Kd7 45.Rc7+ Rxc7 46.Bxe7 Rc6 47.b7 Kc7. But Black has reasonable winning chances after [B] 44...Kd7 45.Bxe7 Kxc6 46.Bxf6 Rxb6 47.Bxg5 Rb2 48.Bxh4 Rxg2 49.Ke4. Better than the game contuation? Let's see...

Sticking with the plan. But it is always a good idea to keep a weather eye out for opportunities that can appear 'against the run of play'. The fact is, Black ought not to have exchanged bishops, as this rook endgame is quite a straightforward win for White. This is one instance in which there are wider issues than to place the rook behind the passed pawn. Had Black played 45...Kd8 or ...Kb8 then 46.Rc6! protects the b-pawn and attacks the f-pawn at the same time.

48.b7 looks like an over-finesse to me. At once 48.Kc4 gains a tempo if Black takes on b6: 48.Kc4 Kxb6 49.Kd5 and you can paint the black pawns white and call them pallid with fear. Has Black any option but to take? None that I can see!

I just discovered that according to the endgame tablebase, my analysis is incorrect; after 49. f4 gxf4 50. Rxf4 Rg7 51. Rg4 Black has a surprising draw with 51... Rxg4! White can walk the king to h5, but Black plays ...h3 and White's extra pawn is no good as it will be on the rook's file.

A third option is 52...Ra7, threatening to check the WK away from his rook. Naturally, 53.fxg5 won't do on account of 53...Ra4+ 54.K moves Kxf5 and Black wins. If 53.Rb5 (or Rc5) ...Ra4+ 54.Ke3 Rxf4 and White has to fight for the draw. And what if 53.Rd5, to cover the check by interposing at d4? 53.Rd5 f5!! 54.Kd4 Ra4+ 55.Kc5 Ra5+ etc. Incidentally, when manoeuvring a rook in order to check from a flank like this, it is almost always good policy, as here, to move as far to the flank as possible - to the a-file, for preference. That is why 52...Rb7? is not such a good option, as in that last line, 55...Rb5+ would not be available. The move 52...Rg7 is much too passive in the light of these more aggressive and effectual options.

After 54...g4 White could win a pawn (55.Rh5 gxh3 56.gxh5 ... 57.Rxh4) but his winning chances would be close to nil. By the way, White ought (see my previous note) to have played the rook all the way to the a-file. "Checking distance" is a useful concept in rook endings. Then 54...g4 would not be an option: 54...g4? 55.Ra6+ Kf7 56.Ra7+ Kf6 57.Rxg7 Kxg7 58.hxg4 Kg6 59.Kf4 Kf6 60.g5+ Kg6 61.Kg4 (+-).

(!!) A very fine move. Aren't endgames wonderful?

This seems so ... crude ... following the previous move!   ----->   I like the look of 57.Rc7+ with the idea 57...Kf8 58.Kf6!! winning a rook. Of course, Black can avoid that by 57...Ke8 58.Rc8+ Kf7 59.Rxg8 Kxg8 but then White just plucks the berries and wins easily. Mind you, the corpus is just as delicti whether you surgically remove the vital signs, or whang him over the bonce with a bludgeon.