If I were to use a single word to explain why I love Zelda, it's exploration. Or freedom. The freedom to explore, kind of. Note that this goes mostly towards the 3D Zelda games, as I truly love them, whereas the 2D games never really amazed me that much.
I absolutely love mocking about, exploring. Every time I walk across an area in a game, I see loads of different opportunities for hidden paths, hidden objects or hidden enemies. But pretty much none of those hidden paths are ever used. The game wants me to move through that area, it doesn't want you to mock about.
The same goes for enemies. There's the obvious, effective way to kill an enemy, and there's the retarded crappy, ludicrously hard ways to kill them. If I have the choice, I never choose the effective way. Unfortunately, few games really have different ways to kill enemies.
That's what makes Zelda so interesting. There's tons of hidden stuff to explore, and there are loads of ways to kill the enemies.
I remember one place from Twilight Princess incredibly well. On Hyrule Field, you fire a bomb arrow up onto a hill to open a way. Running up there on Epona, you encounter some strange marks on the wall. Well, fiddlesticks, there was nothing here after all!
*A couple of temples later, you find a spinner*. Aha! So that's what the silly marks on the wall were. I have to go there! So I head up there, and jump with the spinner in a series of ways that took me ages to manage. On the top, there's a heart piece. Upon getting it, the "heart container get" tune plays. That makes me feel bloody fantastic. Forget killing the boss, forget realising the plot, that beats pretty much anything. The game never gave me a clue towards this, I found it purely through exploration and remembering. I feel bloody fantastic every time something similar to this happens, and that feeling lasts for a long time. In fact, such an experience makes the entire rest of the playsession bloody fantastic.
Another thing about Zelda is that it lets me do things the wrong way.
On the Hyrule Field on the northern exit from Kakariko village (Twilight Princess), a large bunch of enemies approach you. From Epona, those enemies would be a piece of cake. But say I jump off Epona, what then? Then they're bloody hard. There's absolutely no in-game reward for defeating them that way, but it's a lot of fun.
So, if I choose to make the game hard, it's hard. If I choose to make it easy, it's easy. It's completely up to my mood. I love that.
Oh, and there's the music. That's a bloody huge part of this. Part of the reason why exploring is so brilliant is because I enjoy just listening to the music. I don't really have to do anything else, I'm perfectly content with just that.
All the other "main" parts of Zelda are great and fun too, but they're really nothing compared to Zelda letting me use my imagination. I can walk where I want, fight how I want and do mostly everything how I want. I guess it sort of is that I can adapt the gameplay to how I want it.