As of now, string theorists have no explanation of why there are three large dimensions as well as time, and the other dimensions are microscopic. Proposals about that have been all over the map.
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As you say, the way string theory requires all these extra dimensions and this comes from certain consistency requirements about how string should behave and so on.
One of the strangest features of string theory is that it requires more than the three spatial dimensions that we see directly in the world around us. That sounds like science fiction, but it is an indisputable outcome of the mathematics of string theory.
Having those extra dimensions and therefore many ways the string can vibrate in many different directions turns out to be the key to being able to describe all the particles that we see.
Technically you need the extra dimensions. At first people didn't like them too much, but they've got a big benefit, which is that the ability of string theory to describe all the elementary particles and their forces along with gravity depends on using the extra dimensions.
String theory envisions a multiverse in which our universe is one slice of bread in a big cosmic loaf. The other slices would be displaced from ours in some extra dimension of space.
If I take the theory as we have it now, literally, I would conclude that extra dimensions really exist. They're part of nature. We don't really know how big they are yet, but we hope to explore that in various ways.
According to String Theory, what appears to be empty space is actually a tumultuous ocean of strings vibrating at the precise frequencies that create the 4 dimensions you and I call height, width, depth and time.
To tell you the truth, I've never met anybody who can envision more than three dimensions. There are some who claim they can, and maybe they can; it's hard to say.
To the extent that we even understand string theory, it may imply a massive number of possible different universes with different laws of physics in each universe, and there may be no way of distinguishing between them or saying why the laws of physics are the way they are. And if I can predict anything, then I haven't explained anything.
The most important single thing about string theory is that it's a highly mathematical theory, and the mathematics holds together in a very tight and consistent way. It contains in its basic structure both quantum mechanics and the theory of gravity. That's big news.
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