I take issue with using the tesseract in picturing extra dimensions; the hypercube in this case is usually depicted as a cube within a cube, and I think that's unnecessarily complicating the issue. What we're required to imagine in such a case is a cube, extended in one extra-dimension, and then returning to its 3d state. It's so much easier to represent the entire added dimension, instead of just a hint of it, to wit:
Forget the inner cube. Imagine a transparent cube -- say, built of plexiglass, and imagine its centermost point. Like the center, or "core" of a Rubik's Cube, around which everything else turns. Now, connect all eight corners of the cube to that point, and you have six pyramid-like shapes turned inward on one another. The 'floors' of these pyramids form the six outer planes of the standard (3d) cube.
IAW superstrings, "strings" might be considered the lines from the corners of the outer cube to the central point, the "membranes" could be considered all the planar surfaces, and ten dimensions might be considered the eight points of the traditional cube, plus one inner (the 'core,' dimensions fully reduced) and one outer (the whole cube, or all dimensional) points. Depending on how you prioritize the resultant points, rods, and planes, you can arrive at 26 (& many other values) "dimensions."
Inflation putatively occur only in some dimensions, hence the "curled up" theory. Otherwise, we should eventually see these other dimensions 'surface' (unavoidable pun).
Actually, I think a lot of these ideas represent just how far our scientists still have to go. Much of it seems like grasping at straws... trying to make reality conform to our limited number crunching abilities.
It's the luminiferous aether all over again.