I don't understand expected puzzle solutions

You want to look for the best move. If its mate, perfect. Otherwise its about finding the best move. Could be winning a queen, getting a free piece, getting into a winning endgame. Looking for 1. Checks, 2. Captures, 3. Attacks helps a lot with searching for the outcome.

No, puzzles don't have to end in checkmate. The goal of puzzles is to find the best move in the position. Often, there are tactical patterns to look out for, such as pins or forks, that will lead to material gain. It's not always about how to win, some puzzles are even positions where you have to find the only drawing move.

Many puzzles focus on specific tactical motifs such as forks, pins, skewers, discovered attacks, double attacks and more. Your task is to find the best tactical move to gain material advantage, create threats or exploit weaknesses in your opponent's position.