This buys time for castling. Black will also keep an eye on g7 in order to keep the White King boxed in. Worse was 12...Qg7+ when the Queen has fewer options for giving check if the White King moves to the middle.
13. Kc1d2
I was unclear as to whether 13 Kg8 Qg7 was checkmate due to the absolute pin on Black's bishop. The matter didn't come up, so all that mattered was that White thought it was checkmate.
13... O-O
Finally! With this move Black cements his advantage. White has no way to meet all of Black's possible threats which include ...Qc3+, ...Bf4+, and ...Rd8+.
14. Ng1f3
This move protects after 14...Rd8+ 15 Ke1 when the Knight can play to d2 and White's pieces huddle around his King. This move does not address White's weak queenside dark squares. The only way for White to solve that problem is to try to force trades, which will probably require loss of material.
14... Qa3c3
Preventing 15 Ke1 when Black's attack fizzles out.
15. Kd2c1 Be5f4 16. Nf3d2
?? Allows checkmate. Better was 16 Kb1 when White can fight on to attempt trading down into the endgame to minimize Black's advantages.
16... Qc3a1
The holes in White's defense caused by 9 dxe4 were his downfall. Note that while the threat of Black's kingside Rook helped produce mate, actual checkmate is given by only two pieces - the ones that operate on the dark squares.
