Good line.  Black has one more move -- 2. ... Be5, but then it is Bxe5#

And that is much more natural than the two moves by the king allowing black to promote. 

Also,  what if black promoted to a knight instead of a Queen?  (Well, there's still the Bc6+ move suggested by wheelsonfire.)

EDIT:  But Whoops!! as E-Kuko points out, we both missed the black response to 2. Rf3  ...  advancing the pawn to e5 --

So now the question "What if black promoted to a knight instead of Queen?" becomes relevant -- as does the question "What if black plays 1. ... Kxe4 instead of promoting his pawn?"  I have to try these out, but they don't look like the dramatic mates in the solution.

The latter question is pretty easily answered:
1. Ke2      Kxe4
2. Bd3+    Kd4
3. Rxe6+   Be5
4. Bxe5#

but 1. Ke2  f1=N+ provides, I think, a stickier problem 
White has six legal moves:  Kxf1, Kd1, Kf3, Kd3, Nf2, and Rf2+
Black has several interesting responses to each of these. -- I just don't see any easy solution.