Technically, Hammonds postulate only describes the geometric structure of a chemical reaction.  However, Hammonds postulate indirectly gives information about the rate, kinetics, and activation energy of reactions. Hence, it gives a theoretical basis for the understanding the Bell-Evans-Polanyi principle, which describes the experimental observation that the enthalpy change and rate of similar reactions were usually correlated.
The relationship between Hammond's postulate and the BEP principle can be understood by considering a S1 reaction.  Although two transition states occur during a S1 reaction (dissociation of the leaving group and then attack by the nucleophile), the dissociation of the leaving group is almost always the rate-determining step.  Hence, the activation energy and therefore rate of the reaction will depend only upon the dissociation step.
First, consider the reaction at secondary and tertiary carbons.  As the BEP principle notes, experimentally S1 reactions at tertiary carbons are faster than at secondary carbons.  Therefore, by definition, the transition state for tertiary reactions will be at a lower energy than for secondary reactions.  However, the BEP principle cannot justify why the energy is lower.
Using Hammond's postulate, the lower energy of the tertiary transition state means that its structure is relatively closer to its reactants R(tertiary)-X than to the carbocation "product" when compared to the secondary case.  Thus, the tertiary transition state will be more geometrically similar to the R(tertiary)-X reactants than the secondary transition state is to its R(secondary)-X reactants.  Hence, if the tertiary transition state is close in structure to the (low energy) reactants, then it will also be lower in energy because structure determines energy.  Likewise, if the secondary transition state is more similar to the (high energy) carbocation "product," then it will be higher in energy.