Play lots of puzzle games and analyze their design. If the game has a level editor, spend some time with it and see how it feels to work with the elements in it. English Country Tune and Stephen's Sausage Roll were recommended above. I'd second those, and add
Deadly Rooms of Death. DROD is pretty much the pinnacle of puzzle games in my opinion, and features a capable level editor with a very active community behind it.
Snakebird and
The Witness are also highly recommended.
Jelly no Puzzle is also worth a look.
I'm working on building a DROD hold now, and I'm finding that it's a process that can't really be forced. If I sit down to design a puzzle without some sort of inspiration, nothing productive tends to happen. What I do is keep a text file around where I jot down ideas for puzzle design any time they pop into my head, so that when I sit down to build something, I have a bunch of ideas ready to pull from. If the well runs dry, it seems like I just have to wait it out - inspiration can be encouraged to happen by keeping the project in mind, but it still has to happen on its own without being forced.