Everybody loves a good story and in most modern media a story is told in a linear fashion. It doesn't matter if you read a book, watch a movie or a good TV show, you get the information in a supposed order. While, there's nothing wrong with that and also ensures surprises and cliffhangers, there's another way of storytelling only possible in interactive media like video games. Like Sherlock Holmes you're diving right into the situation and piecing together the giant puzzle of the story. And this is how story telling should work in a video game.
Metroid Prime is a nice example for this. It's a game, where you fully explore a world similar to the gameplay of Zelda. It has a very deep and interesting story, yet there are no major cutscenes in the game. Most of the story is told via scanning the environment. You get log book information from scanning lores, computer panels, enemies and more. And piece after piece you puzzle together the magnificent story around the Phazon. You're entering this foreign environment and piecing together the clues all by yourself. It's a much deeper experience than any movie could ever deliver.
Of course in Zelda you can't scan things. Still Zelda offers many different similar storytelling devices. You talk to people, read books in houses, decipher inscriptions on ancient stone tablets or listen to Gossip Stones. Especially the Nintendo 64 Zelda games, Ocarina of Time and Majora's Mask, did a great job with that. One of the reasons, why people enjoy OoT's Hyrule or Termina's Clocktown so much, is because they really let you dive into these worlds. There are many secrets to discover, which makes these game worlds so much more interesting, alive and fascinating. And the information you get from the people and Gossip Stones stimulize the player's imagination in ways, a cutscene couldn't ever deliver. It's like the difference between watching a movie and actually reading the book. But even better, because it's an interactive experience, where you discover the information by yourself.
The storytelling in Twilight Princess on the other hand relies almost entirely on cutscenes. When you're in Hyrule Castle Town you can't talk with most of the people there and the few responsive individuals don't have anything interesting to say. It's boring. There's no story here. On the other hand you ran from one scripted sequence to the next, it's the most cutscene-heavy Zelda game next to Skyward Sword. An experience that is almost movie-like and I don't mean that in a good sense. In Twilight Princess you get to watch little movies about Midna, the Twili and whatever, one after another in a linear fashion. It's just right into your face, you don't have to do anything except finishing the main quests. While in the N64 Zeldas you discover all the secrets by yourself, like about the Sheikah or how the Gerudo society works. There's no big cutscene telling you everything about the Gerudo on your way to the desert. You learn this stuff by yourself, which makes it such a richer and more satisfying experience.
Another nice example would be the stories by grandma in the Stock Pot Inn. While technically they are "cutscenes", it's something you discover all by yourself. And something you might miss. But this is what it makes so rewarding. The Skull Kid and the Carneval of Time are all part of the mysteries in the game and you just found some important pieces of the puzzle. Games are about winning or losing and getting the rewards or failing to get them. The complete story is your reward and it shouldn't be handed to you on a silver platter.
But again this is all part of the giant linearity problem. The developers at Nintendo are way too much afraid about their customers. They can't handle this, they can't handle that, that is too hard, this must be made easier, let's put in Super Guides and and make everything linear. Non-linear world design is way too confusing for the tiny brains of Nintendo's customers. That way we make also sure, that they don't miss any of our genious content. - It's the same with the story, it would be such a shame if you missed any parts of it, so they make sure to shove it all right into your face.
And cutscenes are always game stoppers. They take away the control from the player, which is always bad in a game, which is supposed to be an interactive experience. If I want to watch a movie, I wouldn't be playing a game. You should always have the feeling that you're in control and not just part of some script.
I was actually excited, when I learned that Skyward Sword brought back the Gossip Stones, because on the N64 they were always the source for some of the most interesting details. Too bad that most of the information you get from them in Skyward Sword is about what you can win at minigames. Skyward Sword definitely follows the linear movie-like pattern, the game has many cutscenes and only little to explore. And say, what fascinated you more? Hyrule and Termina on the N64 or the world of Skyloft?
Explorative storytelling is part of what made the classic Zelda games so much more magical. And it's one of the things Zelda needs to get back in order to return to former glories.