I would like to know the physical meaning of the Legendre transformation, if there is any? I've used it in thermodynamics and mechanics and it seemed only a change of coordinates?
In theoretical physics, the basic or defining mathematical properties of the Legendre transformation are used to switch between one form of the energy - or "potential", as the generalized energies are called in thermodynamics - to another.
This is important to switch between the Lagrangian in abstract mechanics that depends on $x,v$ (positions and velocities) to the Hamiltonian, the true energy that depends on $x,p$.
In thermodynamics, the number of applications and "types of switches" is even higher. You may go from energy to enthalpy or Helmholtz free energy or Gibbs free energy by Legendre-transforming with respect to various variables. The transform goes back and forth. As the Wikipedia example explains, there are other useful variables that you may Legendre-transform with respect to, including the charge and voltage.
You may consider the Legendre transformation to be a "mere" redefinition of variables - but that's why it's so important in practice. In reality, the different ways to describe the system that differ by a Legendre transformation are "equally fundamental" or "equally natural" so it's often useful to be familiar with all of them and to know what is the relationship between them. The relationship is given by the Legendre transformation.
It isn't "physical" intuition per-se, but I find the convex-analysis interpretation of the Legendre transform to be the most enlightening.
A convex set is uniquely determined by it's supporting hyperplanes, and the Legendre transform is an encoding of the convex hull of a function's epigraph in terms of it's supporting hyperplanes. If the function is convex and differentiable, then the supporting hyperplanes correspond to the derivative at each point, so the Legendre transform is a reencoding of a function's information in terms of it's derivative.
Here are a few links that illustrate the concept:
http://jmanton.wordpress.com/2010/11/21/introduction-to-the-legendre-transform/ (great in-depth explanation)
http://www.mia.uni-saarland.de/Teaching/NAIA07/naia07_h3_slides.pdf (compuational perspective)
http://maze5.net/?page_id=733 (graphical/visual explanation on my website. Not that advanced, but but with a lot of pictures)