Rob, there are countless books on the issue. Try Polcinski's String theory, or Clifford Johnson's book.The latter deals beautifully with compactifications.

If you want to picture it, roughly speaking, imagine a plane tangent to a sphere, than make the sphere very very small. This is simple to picture. The difficult thing to picture is that at every point in your spacetime you have such a plane tangent to a microscopic sphere.