In this video http://www.youtube.com/watch?v=yzMEAkI-yrQ&NR=1 explaining string theory in a layman way, (a) how did scientisist come to believe that sub-atomic particles behave like strings in the first place? (b) How did scientists leap to believing that these strings can grow into membranes (or, branes) of astronomical proportion? (c) Last, the video explains why there may be 10 dimensions, but instantly start talking about 11 dimensions. Where did the 11th dimension come from? Does that really mean strings move in 7 additional dimensions, and not just 6 additional dimensions, beyond the four we know?
Be careful - superstring theory (or M-theory) is extensions of General Relativity and the Standard model. It is an attempt to extend the successful “merging” of the electroweak and strong forces to include gravity. There is no accepted view that reality actually has 11 dimensions.
Currently electrons and quarks are considered point particles (0-D). String theory posits that what they actually are fields tightly bound up in N dimension. Originally N was 1 (a line or string) since a vibrating string contains energies that can exist in discreet levels much like a plucked guitar string. The math however failed to address certain real world results but if you extend your thinking from N=1 to N=m where m is anything (so electrons could be a rolled up sheet twisted into a doughnut shape) then things begin to fall into place.
More interesting is that when particles interact the current models have them interacting at zero distance. M –theory goes and insists on particles having some dimension, the math insists it is 11 dimension (10 spatial, 1 temporal). Well it insists on 10 spatial dimensions if you want matter to exist. If you just want forces to exist it takes 26.
So where’s the problem? Consider how many possible ways there might be to “roll up” 6 extra dimension to result in our normal 3D world. And which way is correct?
For a really good overview go to this web site. It covers basic and advanced concepts in a way that even I get.
Here’s a recent thread that goes somewhat into the issue of spacetime dimensions as pertains to string theory. As for branes, think of them as a logical extension of the string concept – strings are 1d branes, essentially, and the string theories seem to be able to accommodate higher dimensional versions, as well. I believe that in M-theory, where the number 11 for the spacetime dimensions comes from, each string (in a 10 dimensional superstring theory) is actually generalized to a two-dimensional brane. However, branes are not exclusively motivated through string theory, the idea that we ‘live’ on some brane world moving through a higher-dimensional ‘bulk’ universe arose somewhat independently (I think) to solve the problem that gravity is so much weaker as a force than the other three, which is a major obstacle to finding a unified theory of everything – essentially, the idea is that gravity ‘leaks off’ the brane into the bulk, and that we only ‘see’ the effects limited to our particular slice of this higher-dimensional reality.
Also in the thread above, I link to the full NOVA programme the YouTube video is taken from; if you’ve got the time, it provides a good popular overview of string theory and related concepts, including how the string idea arose (chapter ‘The Birth of String Theory’).
Besides M-theory (not Theory 
)is only 1 candidate for merging quantum mechanics and GR. There’s also Loop Quantum Gravity, Twistors, String-Twistor theory, Causal dynamical triangulation (which had a call SA article a few years back) and others.
Through “though experiments.” Einstein used them a lot.
You can kind of think of it as a game. A few physicists get together and instead of playing charades they play “thought games.”
One poster here said “Currently electrons and quarks are considered point particles”
The game could be "OK suppose they are NOT point particles, what else could they be?
Than the scientists throw out things and come up with answers. So are realistic and some are way off the mark, but eventually someone may thow out an answer and someone says “Wow you may be on to something.”
As Grey noted current Quantum Mechanic formulations have fundamental particles as point particles (0 dimension). When they tried to merge QM with GR doing the math led to ridiculous answers (infinities started popping up, probabilities greater than “1” and so on).
I think someone figured a way to avoid some of these pernicious effects was to give these zero dimension particles some dimension (albeit really, really small). Seems to me the logical step from 0 dimension is to suppose one dimension (a line or string if you prefer). Go from there.
Actually, from what I understand, the string model was originally intended just to unify the Strong and Weak Nuclear Forces (what’s referred to as the “Standard Model” of particle physics), but then they discovered that any string model adequate for explaining the standard model also predicts the existence of massless spin-2 particles, which are exactly what gravitons (if they exist) need to be.