The best way I can describe the story is "layered".
There's at least 3 levels to the story... you'll experience the most immediate, and you can choose to dig deeper to uncover more, depending on how you want to play the game.
The world of Aquaria is a fantasy underwater world, so there is a system of "magic" that Naija uses... being able to sing to change the world around her and to change into different forms. That's one of the main supernatural elements and its also a major part of the story.
There are definitely strong emotional elements. How they come to the surface will be different for each player. Certainly the voice acting conveys a lot of the emotion of the story events, but there is also emotion conveyed through the music and gameplay.