There are a fair number of lines toward the end that make the fact that mate is forced a little hard to spot, but it's certainly doable. A couple example lines:
[FEN "r1bqk2r/ppp3Bp/2n5/8/1b2p3/1P2n1P1/P4PBP/RN1Q1KNR w kq - 0 0"]
1.Ke2 (1.fxe3 Qxd1+ 2.Kf2 Qe1#) Qxd1+ 2.Kxe3 Qd3+ 3.Kf4 Bd6+ 4.Kg5 Qd5+ 5.Kh6 Bg4 6.Bxh8 (6.Be5 Qxe5 7.Bxe4 Bf8#) Qh5+ 7.Kg7 Qg6#
As for learning to spot these mates yourself, part of the trick is recognizing that Black's pieces are hugely more active than White's, especially after the White Queen is gone. Black should look to control the White King's movement easily with the two Bishops and the Queen, and mate is surely inevitable. Finding the fastest mate is always nice, but at the end of the day, any line will do as long as it doesn't provide an opportunity for White to escape!
When calculating mating lines (or any combination), it's easiest to look for forcing moves: moves which leave the opponent with only a single move, if possible. This naturally limits the number of lines you have to calculate. Good candidates for forcing moves are checks, or other threats which must be dealt with immediately. For instance, I found the line above by simply evaluating, at each step, the move that gives White as few resulting choices as possible. Even
5...Bg4 is a forcing move, because it threatens immediate mate via