The first move is easy to see. It puts black in a difficult situation. Black wants to get his King to a2 to protect his advancing b-pawn, but he cannot do so after Bd5. He advances the pawn to b2, but now cannot move his king to a2. If he moves the King to a2 first, the b-pawn is pinned; white just temporizes (2. Bc4, he can move the bishop between those two squares until it is a draw by repetitions).
So, Black moves the pawn to b2. and now white can move his king to c2, preventing promotion at b1. Black's two viable moves now are e4 or d3+
He plays d3+. In either case, white moves his King to b1, finally blocking promotion there. Now black advances his pawn to d2, threatening game-winning promotion at d1. But white has a clever response.
4. Bb3! prevents promotion at d1, BUT offers the bishop to sacrifice:
4. ... Kxb3 -- STALEMATE (I'm happy to say I got this move, but it took a long time - finally, when I could find nothing else to save the day, I saw how this would work.)
On the 5th move I wanted to move the Bishop right to d1, but daily puzzle insisted it should be 5. Bc2. I am not sure that 5.Bd1 was "wrong," but there may be some delicate counting whereby black could have gotten his King to d3 or a3 and white unable to prevent promotion.
The 6th move, King capturing the abandoned b-pawn was a natural choice, but 7. Ba4 was way over my level. I again wanted to move my Bishop to d1. Once again, the purpose is to prevent the black King from getting to the second rank to protect a promoting pawn. The 8th move,
8.Bb5+ was aimed more at keeping the black King off the e2 square than merely checking him. I missed this one also and only understand it in retrospect.
The final clincher, 9. Kc2 was easy to see. Black was forced to abandon his d-pawn, and his chance for winning is clearly gone,
The Bishop moves to a4 and to b5 were way beyond me, but they clearly worked. I suspect that the Bd1 moves that I wanted to play would have failed to prevent promotion and ultimate victory for Black, but I will have to be shown just why.
this is difficult! good puzzle today.