I expected Qa4, followed by Nxc7 - Rxd1+, and recapturing with the bishop means white wins material. But you also have that after pawn push to a4. I do not like this exercise, so please somebody explain what I am missing here.

The black queen would not take the white pawn on a2 in move 4 since the white queen will recapture. If the black queen goes to a3 instead, this is what might happen.

1. ... Bc7
2. Be2 Qa4
3. Nxc7 Rxd1+
4. Bxd1 Qa3
5. Nxe6 fxe6
6. Qxe5+ Kc8
7. Qxe6+ Kc7
White soon promotes a pawn and wins when the second queen delivers checkmate.

If the black queen takes the pawn on c3, this is the outcome.
1. ... Bc7
2. Be2 Qa4
3. Nxc7 Rxd1+
4. Bxd1 Qa3
5. Nxe6 Qxc3+
6. Qd2 Qxd2+
7. Kxd2 fxe6
8. f7 Kc7
9. f8=Q

Either way, white promotes and wins because black has nothing but pawns left.

Black's best option is to lose the queen because black still has a rook, bishop, a knight, and six pawns.

Huh. 4 1/2 years later, I still got it wrong!