How to understand energy function in thin plate splines?

guys! I have carried out some experiments using matlab warp tools from here. But the results are not good, so I decide to read its original paper Principal Warps: Thin-Plate Splines and the Decomposition of Deformations from here to find reasons. However, I am stuck to part II.....

It says the energy function $\mathbf{I_f}$ is proportional to $\mathbf{WKW^T = V(L^{-1}_nKL^{-1}_n)V^T}$, then, I can not understand the whole first paragraph of part II:

Why does the pattern of y-displacements represent an expansion of lengths along y-axis? Actually, I know energy function, but I can not understand its relations to $\mathbf{WKW^T}$, then the pattern of displacements, and then the actions around x or y axis. So, I ask this question about energy function. Thanks a lot!