If you know anything about thermodynamics, there is an application of the Legendre transform in developing thermodynamics "from minimal axioms". I found this very illuminating in terms of intuition.

Specifically, the Legendre transform generates the various thermodynamic potentials (Helmholtz free energy, Gibbs free energy etc.) from energy as a function of extensive quantities (entropy, volume, etc.). In the process it introduces the intensive variables (temperature, pressure, etc.) into the theory. The guiding physical principle behind the optimization problem in this setting is just the second law.