what’s the purpose of the puzzles?

All you have to do is find the best move/moves. Sometimes that's making a checkmate, others are to win material or make a passed pawn or some other thing that will give an advantage. If you don't understand why the solution is what it is you can click the magnifying glass for analysis and see how the computer wants to continue. This should show you how the position is advantageous. 

yeah check the analysis (magnifying glass). Usually the puzzles reach an obvious conclusion but occasionally they end before this. When this happens its probably assuming that you must be able to see a forced conclusion to have chosen the correct moves, which may not be the case if you used the hint or guessed.

There is a specific order which you have to check when you solve these puzzles:

1. Check for checkmate (if you can checkmate immidiately you do not habe to look for hanging pieces)

2. Try to get the queen of your opponent (look if you can trap the queen somehow or if there is eg a knight fork with king and queen possible)

3. Look for possible promotion of a pawn (typical endgame theme)

4. Look for other hanging pieces.

If you keep this order in mind you should succeed well. 

The purpose of puzzles is to tune your brain to following the "rules" of calculation, eg. looking for forcing moves; checks, captures, threats, etc and at the same time exposing you to many themes, patterns and motifs such that you begin to imprint and remember them so that you can identify these much easier in real games. Many of those patterns will be intuitively remembered, if not, you'll at least have built up the practical calculation skills (at least for longer time controls). If you play blitz/bullet, you're relying more on the pattern recognition/memorisation aspects - which do build up over time.

Puzzles are "tactics". You're not always looking for checkmates, often it's just "Find the best move". It's sometimes a winning move, it's sometimes a defensive move to hold an advantage, etc. If there are multiple "good looking" moves - keep calculating both lines until one looks better than the other.

I'd want to add one thing on top of the list of how to solve puzzles (though the easiest ones rarely have this):

1. Is your king safe? If there is an attack against your king, the solution is either to defend against the threat or a counterattack with only forcing moves.

In the more difficult ones there are some where the idea is to save a draw in a lost position either via stalemate or perpetual check.

To make yourself improve