In Stanley's Algebraic Combinatorics, Chapter 2 is spent discussing the application of the Radon transform to determine the eigenvalues and eigenvalues of the adjacency matrix of $C_n$, the $n$-cube. (The definition used in the textbook seems to be the discrete version, $$\Phi_\Gamma f(v)=\sum_{w\in\Gamma}f(v+w),$$ as opposed to the continuous version, but I guess they're basically the same thing.) However, the book only serves to explain the transform within the context of the $n$-cube and not in general situations, which has me somewhat confused.
What is the intuition behind the Radon transform? In particular, what is going on beneath the surface that makes it useful and why does it give us more information regarding certain algebraic structures (e.g. the adjacency matrix of the $n$-cube)?