It looks more like a cube in 3-dimensions, not a cube within a cube. That diagram is not what it would look like projected onto 3-space, it is rather some scheme for conveying information about the shape. See the pictures and animations at: http://en.wikipedia.org/wiki/Tesseract
I'm thinking of that other classic book, Flatland. Picture a cube if you lived in 2-dimensional space. You might see it as a square, or as an oblique slice through a cube. But not as a matrix conveying the facts about a cube.
Or maybe I'm missing something. The idea of projecting 4-space onto 3-space, or 3-space onto 2-space, may not be the correct analogy for perception here. Also, the space itself in which the tesseract or cube or square lives may not be straight. Think of curved space-time. A standard 2 dimensional space is the straight x/y coordinate system, going off to infinity in all directions. But another is the outside surface of a sphere, closed up eventually, but locally looking nearly flat (measure the angles of a triangle and subtract 180 to get the slight curvature). An then there a are distorted versions of each, x/y or sphere.
Really I just want to think of a tesseract as a solid shape I see at one moment in time, followed by another moment and another moment until it is gone. That way time is my 4th dimension. If everything is laid out straight, I guess a one-meter tesseract is a one-meter cube the appears all at once and stays the same until it disappears, after 1/c ( = speed of light) seconds (?). But if it lays at an angle in 4-space, or 4-space is curved, or 4-space is closed, then who knows. I just can't picture it being a cube within a cube. Then again, I feel like I live in Boxland at a moment.
Add to that, the time dimension really does seem to different physically, and 4-space has an infinite number of smooth coordinate structures, not just straight, closed spherical, etc. While 2-space, 3-space, 5-space, 6-space, etc. all have a limited number of structures, 4-space is the exception and has an infinite number.