Manifolds certainly arise in physics most dramatically in general relativity. Even before that they play a rôle in classical physics in the solution of 2D linkages whose configuration space may be a manifold of genus greater than one. This makes it reasonable to investigate manifolds as a possible mechanism to explain particles. String theory studies manifolds embedded in manifolds. The outer manifold is approximately the Cartesian product of a very large manifold and a very small manifold. I say approximately for the Cartesian product of curved manifolds inherits the curvature of its factor spaces—that is when you endow the product space with a metric constructed in the obvious way. The embedded manifold is also small and of lesser dimension than the small manifold.
The small manifold is compact, and indeed very small—probably about Plank’s length. The large manifold is the 4D manifold of general relativity. Susskind does not explain why theorists like the particular dimensionality. The first string theory postulated a string which is a 1D manifold. 1D manifolds are scarcely manifolds but soon 2D manifolds were proposed. These were posited either having boundaries or not. They were embedded in the large manifolds and often wrapped around so they could not shrink to a point, rather like a rubber band around a pencil.
Susskind is very impressed with the fact that our vacuum seems to have a zero energy density. It seems that quantum mechanics insists that virtual bosons contribute a huge positive energy density to vacuum while virtual fermions contribute a huge negative energy density. These two dissimilar sorts of particles need to cancel each other’s effects to one part in 10120 in order for our neighborhood to be livable. I am not impressed. (I am impressed with his other Anthropic arguments however.)
Susskind skips the math that explains how it is that a big bang, from the outside looks like a sphere expanding at the speed of light, while inside it looks like an infinite space. Here is the math.