The reason the whole idea of 11 spacetime dimensions came about is because the equations of string theory without these extra dimensions have "quantum anomalies"...namely, the creation and destruction of energy, which is obviously kind of a problem. But, with 11 spacetime dimensions...voila! Problem solved.
Then, of course, people were like, "but why don't we experience these extra dimensions?" And string theory physicists responded with the Kaluza-Klein idea...in a nutshell, they're curled up so small we can't see them. Which is kind of hard to picture$^1$, but it explained it.
So, then, what would constitute as experimental evidence of these extra dimensions? There are three things that would provide evidence (according to Brian Greene's book The Hidden Reality, page 94-95):
- Gravity: Basically, because space is the medium for gravity, more dimensions provide more space for gravity to spread through. The idea is that the strength of gravity becomes diluted as it spreads through the additional dimensions (this offers an explanation of why gravity is so weak). If we could measure gravity's strength over distances smaller than the extra dimensions, we should find it's strength to be stronger. The problem? Measurements on scales as short as a micron haven't found any deviation.
- Missing Energy: If the extra dimensions exist and are much smaller than a micron, they will be "inaccessible to experiments that directly measure gravity's strength." The LHC provides another route, namely looking at collisions between speedy protons and the debris left behind. Why? The debris could be squeezed into one of the tiny dimensions, thereby carrying away energy, which would be recorded.
- Mini Black Holes: Anything can become a black hole if compressed sufficiently. So, if there's extra dimensions that result in gravity being stronger when acting over short distances it should be easier to create baby black holes (i.e., two protons slammed together with enough force might be able to create a tiny black hole) which would then disintegrate into particles that could be recorded.
Again, it should be noted that none of this has been found, though scientists are looking for it.
Hope this helps!
$^1$It might be easier to picture this idea if you think of a really, really, really tall straw (taller than the Empire State Building). Up close, you can see it has all three dimensions, right? But from a distance, it just looks one dimensional. Now imagine that straw was smaller in diameter than an atom's nucleus...