Why do we say that a transition state that resembles a stable species in free energy also resembles it in structure?
EDIT for a different approach:

The energy of a quantum mechanical system of nuclei and electrons is governed by the positions of the pieces. So, we're looking at the energy associated with the wave functions of all the pieces.

For two species that are similar in structure, the idea is that those two structures lead to relatively similar solutions of the wave functions. If the wave functions are similar, the energies of the particles described by those wave functions must be similar as well.