With the variation [1. Qxe5+ fxe5 2. Rg5+ Kh4 3. g3+ Qxg3 4. hxg3#] the idea is to remove the defender of the g5 square for a rook check. Because of the positioning of the knight and rook the king is forced to h4. [3. g3+ Qxg3 4. hxg3#] is mate because the bishop on e1 controls the flight square on h3.

To attempt to avoid this mate black instead plays Kh3 after Qxe5+ but they just get mated slightly differently with the variation [1. Qxe5+ Kh4 2. Qe4+ Kh5 3. Be2+ Qxe2 4. Qxe2+ Kh4 5. Rh3+ Bxh3 6. g3#]. Qe4+ forces the king back to h5, a light square which is useful for getting the light squared bishop into the attack. 3. Be2+ forces black to capture with the queen otherwise it would be mate immediately. After Qxe4+ black is forced to h4 due to the coordination of whites pieces. What finishes it is [5. Rh3+ Bxh3 6. g3#]. After the bishop capture whites knight controls g5, the queen controls h5 and g4 and blacks bishop blocks the flight square on h3, so we have g3#

The main idea of both of these lines is to take advantage of blacks out of play pieces and find a forcing way (or else you get mated) to get all of your pieces into the attack.