Existence of extra dimensions is predicted by the string theory. The string theory is one proposed 'theory of everything'; a theory that would unite the going-ons of the smallest scales (quantum mechanics) with the macroscopic scales (general relativity). It's the Holy Grail of physics today, and without such a theory we can't explain what happens when a lot of matter is packed in a very small space − the singularity of a black hole or the beginning of the big bang (hence the reference in this thread's title, I assume).
String theory says that the smallest building blocks of everything are little 'strings' (around the Planck lenght or 1.6 × 10
-35 m) that can vibrate in different ways. Depending on the manner of vibration they can manifest themselves as different elementary particles, like photons (light) or quarks, which combine to form protons, which combine with neutrons to create atoms, which create macroscopic stuff that we know.
Now about dimensions. Here is an illustration, courtesy of professor Richard Wolfson:
http://i35.tinypic.com/1zpmzqw.png
In the upper part, you are looking from afar, and you see an ant that appears to be able to travel on a stem in one dimension only. When you move in close enough, you see that there is actually another dimension. That dimension is 'curled up on itself', in a kind of sphere. You couldn't see it before, because you were looking from a great distance.
Now, remember that these strings in the string theory are small as fuck. They operate at Planck length, the smallest physically possible length. What if there are more dimensions curled up there, and we simply aren't close enough to that tiny scale for them to have any effect on us? Of course, we can't really imagine another dimension very well, so to us it's just extremely complex mathematics, but for subatomic particles created by the strings, like the protons that they will be colliding in LHC, it may be a different story. It's also theorised that the hypothetical graviton particle, which supposedly 'carries' gravity, 'leaks' into these curled-up dimensions (since it's small enough to access them), which is why gravity is such a weak force and only becomes considerable when dealing with massive objects like planets.
I know this is not very clear. Since I don't really know what I'm talking about, I can't explain it very well, but hopefully this gives you at least some idea.
