In this position, the king doesn't have enough pieces to defend itself (Only the pawns are defending the king no other pieces). Also, the a2-g2 diagonal is covered by the bishop on b3, so the king cannot escape to the g8 square. Finally, the knight tries to open up the h-file by putting itself at the g6 square.
In this position, the g7 pawn is blocking the king from escaping. If the king was on the h7 square it would still be the same thing because of the g6 pawn.
Now I will show you a some games that involve the Greco's mate
This type of checkmate is delivered by a queen, but it doesn't matter if the checkmate was delivered by a queen or a rook it is still the Greco's Mate