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A lot of this is just tactical intuition and pattern recognition, which will improve as you get more experience playing and solving puzzles. This is also a tough puzzle that combines a few different motifs and demands some nontrival calculation, so don't feel too bad about missing it.
Even without an obvious mate, the awkward position of the Black king is a hint that you should look carefully at the different forcing moves at your disposal. Notice that the Black rook blocks g8, so after 1. Ng5+ hxg5 2. Rh3+ Kg6 the king has to come out into the open amid several aggressive enemy pieces, which often leads to a decisive king hunt.
Keep in mind too that in solving the puzzle you don't really think "hey, I can win the queen" right from the beginning, but rather you notice that 3. f4 threatens mate in a way that Black can only prevent by sacrificing the queen.
Solving problems are simpler than finding the tactic in a real game because you know it is there. Many (not all) tactics start with a forcing move: a check, capture, or mate in 1. So, look at all these posibilities first. Ng5 is a check. If the King moves back to h8, the N forks on f7. So, hxg5 is forced. This opens the h file which looks good since the king is there. Rh3 is a check, another forcing move. The king is forced to g6. f4 is also a forcing move because it threatens mate in 1 (f5). To prevent f5, black is forced to gxf4 (Qd7 avoids the mate at the price of the queen. It also results in mate, but it is a ways off). Now Qxf4 threatens mate in 1: Qf5. (why not Rxf4: generally, look at moves that place the queen closer to the enemy King first). Now there are only some useless moves by black to postpone the mate (and/or lose the queen). How do you see all this: look for *forcing moves*, finding a sequence that leads either to material advantage or to mate. In a real game, one often gives up before you get too deep. Puzzles force you to keep going until you run out of forcing moves, because you know the solution is there somewhere. Some more difficult puzzles have moves that do not appear to force the next move by the opponent, but for which all opponent moves are inadequate to meet the next threat, which is often hard to see. This puzzle is not of the harder variety. Puzzles are good training because the same techique of following forcing moves until they run out is often userful in real games.
Yeah. I immediately considered Ng5+ because I know it's a puzzle and this looks like a puzzle move. Puzzle moves always have a specific threat. This threatens to win the queen if black doesn't take.
Without even thinking what may happen next I can play it because I don't see any other threatening moves. Then Rh3 is the only decent looking move and so on.
But it is a lot about pattern recognition also. This one not so much because black's moves were pretty forced.
In trying to solve today's Daily Puzzle (Finishing Touch), I tried to understand the thinking that led to the first move. To sac a minor piece to restrict the King makes sense if a mate was obvious. I did not see the moves that would lead to the winning of the Queen. How does one analyze a position to even be aware that such potential exists?
Somewhat discouraging.