Marcel Berger states Thom's Principle as:
"rich structures are more numerous in low dimension, and poor structures are more numerous in high dimension."
This is in Geometry II (Springer-Verlag, Berlin, 1987. Google books), pp.39-40, after a discussion of regular polytopes shows that there are only three convex regular polytopes in dimensions larger than 4, but six in dimension 4, five in dimension 3, and an infinite number in dimension 2. He then lists further examples of the principle: e.g., simple Lie groups illustrate rich structure in low dimensions, and topological vector spaces illustrate poor structure in high dimension (all homeomorphic in finite dimensions).
In so far as Thom's principle is true—or at least holds widely—my question is:
Why should low dimensions exhibit richer structure than high dimensions? Is there any generic reason to expect this?
It might also be interesting to track down Thom's own formulation of his principle, to understand the context in which he proposed it.
Edit. Here is a snapshot of Berger's examples (p.40):