White can also win with 1. Qf1 Rh2 2. Qg1+
a) 2. ... Kf3 3. a7 Rh8 4. a8=Q Rxa8 5. Qh1+ skewering the king and rook.
b) 2. ... Kh3 3. a7 and black cannot prevent white from queening.
c) 2. ... Kf4 3. Qxh2+
Other second moves from black allow white to just take the rook as in the solution even 2. ... Rd2 3. Qe1+ forces 3. ... Rf2 and the rook can be taken.
maybe not such a good puzzle - lots of variations
It's a good example of simplifying to a more straightforwardly winning position. Giving up material to simultaneously eliminate black's mating threat and forcing home the pawn promotion is a good practical choice. Even though white can probably win keeping the queen and rook on, e.g. 1. Qc5 defending the mate and threatening the bishop 1. ... Re2 2. Qg1+ Rg2 (otherwise 3. a7 leads to promotion) 3. Qe3+ forking the king and bishop.
how?