Mark Chu-Carroll has a very nice discussion of what "extra dimensions" actually mean in theories like string theory. It's not the same thing that hack SF authors mean when they talk about "dimensions" in which the Nazis won WWII (that's "multiverse theory" or possibly "landscapeology" or possibly "late-night stoner bullshit"):
A better way to explain, but a slightly less intuitive one is to not separate dimensions quite so much. The set of dimensions in a space is the number of pieces of information that you need to identify a unique location in that space.
On a plane, you can put down a coordinate grid, and identify any point on the grid with only two numbers. In a region of three dimensional space, you can specify any location in that space using only three numbers. But getting beyond three, we start to have some trouble.
Try thinking about the colors on your computer screen. Basically, for each pixel on your screen, there are three lights that can be turned on in different brightnesses: one red, one green, one blue. The color of each pixel is determined by the brightnesses of three lights for a pixel: if all of the lights are completely off, the pixel is black; if the blue light is on very bright, the red light is on halfway, and the green light is off, you'll get a bright bluish-violet; if the red light and the green light are on full brightness, and the blue light is on halfway, you'll get a sort of pastel green. So the colors of your computer screen have three dimensions.
Try thinking about the flavors of food. It's commonly asserted that we humans have the ability to perceive sweet, salty, bitter, and sour flavors. Every flavor that we can experience is some combination of sweet, salty, bitter, and sour. Therefore, flavor has four dimensions.
Really getting an intuitive grasp of anything beyond three dimensions is a tough problem. I can't do it myself, but I can appreciate a good attempt at explaining it.
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One thing that I find works well at getting your brain round this sort of stuff is being very bored in a maths class and spending your time drawing hypercubes. Or playing 4D tic-tac-toe with your equally bored friends.
Less boringly, there's a couple of Linux screensavers that draw out various hypershapes and make them rotate a lot. Stare at one for long enough and it aaaaalmost starts to make sense. And then your brain explodes.
Let's face it, all most (I would say any) of us can do is conceive of three spatial dimensions. I have sincere doubts about anyone who says he can visualize more than three. I once heard, probably on Nova or something, a well-known (but not so well known to me that I can remember his name) physicist say that he really had no intuitive understanding of multiple dimensions; he simply (!) could do the math required. It is, as I have said before, as if one could speak a language fluently without understanding it. But this explanation actually makes some kind of sense. If you let go of the need to visualize more than three spatial dimensions, it's easier to understand that a thing could require more than three "pieces of information" to fully describe its properties.
(I am still having to purge cookies sometimes)
That is a nice explanation, I would only add one thing- the 3 numbers that specify location in our space are fundamentally equivalent, even if we happen to live in an asymmetric situation. For example you do not need a separate laws to describe gravity working in the x direction and gravity working in the y direction, even though on earth you are not likely to fall sideways...
The strength of the extra dimension speculation is precisely that these new pieces of data are related to the old. We are in an asymmetric situation (some dimensions are small and some are large, the story goes, or all are large but some are curved as in Lisa's work), but the laws describing both are the same, so for example "falling down" in the extra dimensions turns out to be related to the electromagnetic force. For a theorist that is kind of cool (and of course "cool" is far from "correct", I know).
Nice analogy between red-green-blue in colour-space and length-width-height in space-space. Which makes your CMYK printer four dimensional, and your hexachrome printer 6D.
Personally, I think visualising complex three dimensional shapes isn't trivially easy. 2D shapes like hexagons aren't hard, but can you draw a dodecahedron or buckminsterfullerene molocule (soccer ball) without effort?
In the string theory context, I had the idea of explaining extra dimensions by the effect adding one dimension has on removing mirror assymetry. It's easy to show that a 1D mirror assymetry is removed if you are allowed to rotate in 2D, and a 2D assymetry is removed is you're allowed to flip the paper over in 3D. Likewise, you right and left hands would be identical in 4D, etc.
Mathematically, this continues indefinately, but in string theory, it becomes a valid question to ask, how many extra dimensions do you need before the four forces of nature are identical, if their various manifestations we observe are the result of assymetries in space and time?
Human taste has five flavors: sweet, salty, bitter, sour, and umami. That is what happens when your idiot government "loses" $30 billion in New Orleans and gives Head Start a 20% bigger annual budget than the NSF. Those who discover are those who dominate.
[Two exceptions: massive reproduction to overwhelm rationality by sheer force of numbers - the slavering mob. Social advocacy tantamount to cultural suicide - lethal penitence for sins of productivity.]
Did Witten simplify the concept of extra dimensions with twistor string theory?
By utilizing complex-3D spatial dimensions [degrees of freedom] rather than the M-theory [real-3D + imaginary-3D for a total of six dimensions] which are actually six axes, he reduces the number of dimensions to three.
This is more in consistent with the 1797 demonstration by Caspar Wessel that the existent imaginary unit is more "invisible" than "imaginary". Paul J Nahin [PhD EE, UNH] discusses this in 'An Imaginary Tale' - one unit rotated 90 degrees counterclockwise from origin of the real line [essentially not a degree of freedom].
David Hestenes [physics emeritus, ASU] discusses the imaginary unit in 'Grassmann's Vision' [from Hermann Gunther Grassmann (1809-1877): Visionary Mathematician, Scientist and Neohumanist Scholar, 1996 (Gert Schubring, Ed.), Kluwer Academic Publishers, Dordrecht] and on his website regarding "Universal Geometric Calculus'. Hestenes also wrote 'The Kinematic Origin of Complex Wave Functions' and many other papers.
These pages look at complex functions of a complex variable u+iv = f(x+iy) in various ways. One can project the 4D surface in the same way computer graphics render 3D surfaces. Our eyes and brains have a lot of hardwired machinery evolved to infer 3D structure from 2D images without conscious effort. It is a difficult effort to interpret 4D depth cues. The complex exponential function makes an interesting exercise to interpret the surface, as it's commonly used and has a little bit but not too complicated structure to make sense of from the different projections. Though the projections look like quite different shapes to our eyes interpreting them as 3D surfaces, they all represent rigid rotations of the 4D surface in the same way that a rotating circle may appear as ellipses and a line segment as it rotates.