I'm going to quote somebody, because they said it very well:
<"There is an appeal in Zelda games that goes beyond the gameplay. Part of what gives the atmosphere that magic is the feeling of being a kid (or teenager in some cases like OoT) going out to explore the world and growing through experiences and discoveries. The world is full of vibrant characters that made you want to know more about their backstories">, and there is a feeling that there is so much more than lies beyond the borders that just makes you want to adventure forever. The worlds are just full of life.
Not only that, but the old feature of actually being the character remains. It's lost its place among other titles these days... now you play as a character that's already been developed and has their own sense of things, but with Zelda games, YOU are the hero. Even as Link gains more and more personality, that bit is still there. The hero is you and you are the hero. Link is somebody players can identify with because not only do we become him, but he's just a normal person like we are- he's no ingenious scientist, he's no warrior, he's no reputable hero in his world. He's just a kid that gets caught up with destiny, picks up a sword, and ventures out to save the world with little to no knowledge of combat, the world, or heroism. He learns along the way just as we do.
It's not just the gameplay and the characters and the world, either. The music.... it stands out from the rest. The music flows with the various events of the games and helps to weave intricate stories full of various emotions and experiences. From the booming brass the first time you step into Hyrule Castle in A Link to the Past to the to the delicate strum of Sheik's harp as he speaks in poem in Ocarina of Time, the music allows you to just... feel the very soul of the game- of the legend.
The series stands out from others. You can play another really fantastic, amazing game, but in the end, it's just another generic game. Zelda isn't like that. Zelda strives to be something more. Zelda doesn't focus on mimicking our world as most games these days do, but instead has a world of its own, full of its own races, religion, legends, etc. I guess this is because the series hasn't been abandoned after over 20 years, so it just keeps getting better in every aspect of gaming.
What's funny is that the series can rinse and repeat and still be successful. It's almost always the tale of a hero saving a princess, and even when it's not (LA/MM), the hero/princess thing still makes a small appearance in some way. It never gets old, because it manages to stay the same while ever changing. There's always that feeling of being in a fairytale- dark and gloomy or bright and vibrant, it doesn't matter- that piques my interest every time.
I embrace its cliche because it's not cliche.