Why do we say that a transition state that resembles a stable species in free energy also resembles it in structure? My textbook states," According to the Hammond-Leffler postulate, the transition- state structure for a step that is uphill in energy should show a strong resemblance to the structure of the product of that step."
Also stating,"One way that the postulate can be stated is to say that structure of a transition state resembles the stable species that is nearest to it in free energy. For example, in a highly endergonic step the transition state lies close to the products in free energy,and we assume,therefore,that it resembles the products of that step in structure as well. "
(Source:Solomons' Organic Chemistry
Book by Craig B. Fryhle, Scott A. Snyder, and T. W. Graham Solomons)
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.