1… Qg6+ 2.Kf1 (or Kh1) Ng3+, followed by Qxd3
Note that Ng3+ also covers the e2-square, preventing the king to go to e2 and defend the queen.
1… Qg6+ 2. Kh3 Nf2+3. Kh4 Qg5#
The point os 1… Qg6+ is to force the white king to a more vulnerable square.