You're talking about
brane theory. The problem with being able to enter the
higher dimensional bulk would be that, in spatial dimensions beyond three, gravity and electromagnetism would be much weaker. Electromagnetic fields in three dimensions follow
Coulomb's law: F=k(q1*q2)/r^2. In four spatial dimensions, the electric field would be divided by the distance cubed, which would mean that the electric field would decrease significantly quicker (the same would hold for gravity, which is F=G(m1*m2)/r^2, and I assume it would be the same for the strong nuclear force) - life as we know it would be unable to survive in such conditions. And, of course, this is assuming that physical laws would even be the same in other branes or in the higher dimensional bulk.
As for why we only perceive three dimensions, the answer is simple. Our evolution has only required that we perceive three spatial dimensions and one time dimension. We don't get to perceive things because it's interesting to us, we only get to perceive things that allowed for better survival in our ancestors. Like anthile said, check out
Calabi-Yau manifolds. Even if these things were perceptible (which they aren't, not even by high powered particle colliders, hence why string theory is nothing more than fancy mathematics that proposes something that
could be a theory) we would have had no evolutionary advantage in being able to perceive them.
In addition to string theory, also check out
Octonion physics,
loop quantum gravity,
group field theory, and
supergravity. The point being, string theory is simply the 'grand unification theory' that's had the most publicity. There is very little experimental or observational evidence to back it up, and the idea of requiring 11 dimensions is just coming up with a hypothesis and then attempting to change reality to fit the hypothesis.
