Take it from a Caltech Physics major:
1) String theory at this point is merely speculation. In fact, I'd call it mathematical bananay. In order to provide a shred of evidence for string theory, it'd take a particle collider the size of the solar system.
The only experimentally supported theory at this point is the Standard Model. Anything more than the SM is at best an educated guess (which will soon be proved or disproved with the activation of the LHC) and at worst a bunch of mathematicians getting too excited about what is possible and coming up with a "theory" which we don't have a shot at getting evidence for within the next thousand years. Keep in mind we can't DISPROVE them either as for any theory to be even remotely acceptable, it has to reduce to the currently supported model.
If you've ever seen "The Big Bang Theory" the CBS sitcom, they actually joke about this. When Leonard, PhD argues with his girlfriend, PhD about theories of physics, his girlfriend ends the argument with "BUT WHAT ARE WE GOING TO TEACH THE CHILDREN?!" and breaks up with him over their disagreement. This is a joke on the fact that these advanced theories are almost like religions in their lack of experimental verifiability (I hope this comment doesn't end up getting my post banned).
In fact, a Nobel Prize winning physicst (Sheldon Glashow) left the Harvard Physics department because they had "converted" to string theory.
2) Mathematics only describes what is possible. Just because it is possible doesn't mean it is real. I can envision a universe of infinite dimensions and it will be mathematically reasonable. That doesn't mean it EXISTS, but it is possible. This is because math is simply an ABSTRACTION - a concept and set of tools which are useful.
For example, math can easily describe a universe in which there are only two dimensions. In fact, such a picture is extremely useful in certain situations. That does not mean we live in a 2-D universe.
3) Physics is a science, which means experimental evidence is the last word. In the "Golden Age" of physics in the late 19th and early-mid 20th centuries, the best theorists were every bit the effective experimentalists as well - guys like Faraday, Einstein, and Bohr.
4) As far as the link - assuming 10 dimensions do exist, that is by far the most logical explanation of what they actually mean that I've seen. I've never been able to envision what the extra dimensions in those 26 dimensional theories actually represent. They seemed to me more of a mathematical convenience so the calculations can work out correctly.
Anyways, have fun twisting your minds around extra dimensions
