First, because of their complexity, we have to make thorough observation before thinking about any candidate moves. Rushing out ideas right away will not help us. After these ideas get refuted, our mind will get stuck. We must adopt a systematic approach. By finding out where are our weakest links and where are our strongholds, we can put our focus on the correct areas.
Second, almost all of them are not starting with a check. So there are many more choices for the first move. Any legal move is possible and should be considered. How do we pick out candidate moves which has higher potential to be our solution? This forces us to learn to analyze a position, pinpointing the familiar patterns and distinguishing good moves from useless moves.
Third, because the problems are man-made, the designers have taken great effort to weed out those non-solution moves. We have to test all our candidate moves and remember all the moves we have tried and failed. It trains our memory to remember all variations, sometimes up to more than 10.
Fourth, we have to be patient, double-checking and triple-checking our solutions, going through all possible defenses slowly and finding all finishing checkmates. This is how we think in our tournament games.