1. e4
White moves his pawn from e2 to e4. This serves several purposes. The main one is that it opens up diagonals so that the queen and bishop can get out.
1. ... e5
2. Bc4
Black follows by moving his e7 pawn to e5 for the same reasons, and White moves his bishop to c4.
2. ... Nc6
Black follows up by moving his knight to c6. This helps control th which is a good strategy because pieces in the center control more space than pieces on either side. After controlling the center, he will occupy it and make an attack on White's king.
3. Qh5
White brings his queen out to h5. Bringing your queen out early is normally a big mistake because she can be attacked easily. The Queen will run for shelter or be taken, so the opponent simply brings out his pieces and simultaneously attacks the queen. Soon he has control of so much more space that the first player doesn't have room to maneuver.
3. ... Nf6
Black now brings out his Knight, simultaneously attacking the Queen and White's pawn at e4. He overlooked an important reply, though.
4. Qxf7 mate
The Queen attacks the King. The King cannot move to another square where the Queen would not capture it. The Queen cannot be captured. The King cannot take it because the bishop would then take the king. Nothing can be moved in between the queen and king. The game is over and White has won.