Do you see that this position is structurally similar to the previous one? Again Black’s king has no flight squares; again there is an open diagonal leading toward it (though from below, making it a little harder to see); and again all that prevents White from mating by planting a bishop on that diagonal (Bg7) is Black’s rook. So White sticks his queen flush against the rook with Qf8+, choosing the attacking square that also enables his queen to give check and thus force the issue. Black’s only replies are RxQ or Rg7, permitting White to play Bg7# (or BxRg7#) next move.
The point about giving check can be made more strongly by looking at Black’s own threats at the other end of the board. Black has a queen, two rooks, and a bishop all trained on the White king’s position. If White plays a different attack on Black’s rook—say, Qd8 rather than Qf8+—Black is not forced to take any defensive measures. Instead Black gives a check of his own with Qxg2+, and his checks don’t stop until White is mated three moves later. By giving check himself with Qf8+, White holds the initiative and never gives Black a chance to start his own mating sequence.