A good question. Many philosophers have been discussing this for last last few thousand years, and still have not finalised an answer. If we did not have thought or consciousness, we would not know that we exist. Rene Descartes (from the ideas of Socrates) famously said "I think therefore I am", and even though this should not be taken for proof that we exist in a physical state, it can be taken as proof for our minds existing, as we are able to think and doubt etc. If we use this to help answer your question, we can say that because we have thought and consciousness, we prove our minds exist, however, we might not physically exist (although, I'd say it would be helpful to have a physical existance in which to store one's mind!).

What is apparent, is that we are not able to understand the human brain, because we are inside it, it is only that when we take an outsider's view, that we are able to see the working of it more clearly. Applying this to the reson we exist in this dimension, it could be thought that if we are only able to think in 3 dimensions. If we thought in, say, 8 dimensions, then we would be able to exist in 8 dimensions, it is just that fact that we can't imagine the extra dimensions, that we do not think of ourselves living in them, yet it doesn't mean they are not there.

But maybe this answer was just a figment of your imagination after all...

Although I feel most of the ground is already covered, I like the question and I very much like Heidi's answer. But there is more.

The state in which we live is generally known as a state of uncertainty. Just as in an atom, the electron is in a state of existence that we can not define (Heisenberg uncertainty principle).

Maybe the way to go about this is statistically, the chance that we are, added to the chance that we are not, is always one.

I would also like to add some humour. Philip K. Dick, the science fiction writer, once stated: "Reality is that which, when you stop believing in it, does not go away.

i think  you may be onto something, quantum physics clearly has a role for an observer. i think the universe may be subjective, if you take a thousand people they will all agree that a planet is in such and such a place but there are millions of observers so the probability of it being anywhere else is miniscule. if you then shift to a quantum level then there are less observers and so the the chance of it being where it isn't predicted grows and so you get the quantum weirdness.i think the universe is probability based and as such every observer makes the universe more predictable but we only see the most probable outcome    

There is no reason. The universe doesn't need any reasons.

Just matter and energy, and time, to interact in ways we only measure and try to may sense of.

Chance plays a large part in it.

Possibilities and probabilities play out over the eons.

One possibility, the one that you experience as a living sentient being, is that a brain would evolve to a size that would allow you to become self aware.

And to use the word reason... "a basis or cause, as for some belief, action, fact, or event", and then to attempt to apply that to the cosmos.

Theories about multiple dimensions are only theories.

Theories about multiple universes are only theories.

People exist above and beyond their perception that they exist. Otherwise they would dissapear when they went into deep sleep or a coma. A person's self-awareness (consciousness) is only a subset of their mental and physical being.

As for the universe, of course it exists. Otherwise, how could I be typing this?