What properties make the Legendre transform so useful in physics? The Legendre transform plays a pivotal role in physics in its connecting Lagrangian and Hamiltonian formalisms. This is well-known and has been discussed at length in this site (related threads are e.g. Physical meaning of Legendre transformation, Why is the Hamiltonian the Legendre transform of the Lagrangian?, Equivalence between Hamiltonian and Lagrangian Mechanics, as well as many others).
I am not asking how the Legendre transform is defined, or how it works to connect Lagrangian and Hamiltonian formalisms, so let's take all of this for granted here.
(The main question) Why specifically is the Legendre transform so useful? Starting from a Lagrangian viewpoint, principle of least action and all that, one typically defines the Hamiltonian via Legendre transform and shows that this new formalism is oh so useful for a myriad of reasons.
But why does this happen? What is it about the Legendre transform that makes this possible?
(Geometrical viewpoint) The Legendre transform has a pretty clear geometrical interpretation in the context of convex analysis: it can be used to switch to a "dual" description of a convex set in terms of its supporting hyperplanes (see e.g. the nice description of how this works in this answer, or any introductory book about convex geometry).
Lagrangian and Hamiltonian are also often understood in (differential) geometric terms: one can think of the Lagrangian as a functional on the tangent bundle of some underlying differential manifold, $L:TM\to\mathbb R$, while the Hamiltonian is a functional on the cotangent bundle, $H:T^*M\to\mathbb R$.
Are these geometrical interpretations related in any way? Is there merit in thinking the Hamiltonian as related to the Lagrangian in a similar fashion as how one can describe convex sets in terms of their supporting hyperplanes?
I suppose to some degree this is trivially true: $H$ being the Legendre transform of $L$ with respect to the $\dot q,p$ parameters means (I think) that some sections of the epigraphs of $L$ and $H$ are related in this way.
But that still doesn't tell me why such a dual description of sections of the epigraph of the Lagrangian should be of physical interest..
(Functional viewpoint) Another possible route to an answer might be in asking about the properties of the transform that make it useful. In other words, if I were to start from the Lagrangian formalism, and consider different possible functional transforms $L\mapsto G[L]$, would I get to the conclusion that $G=(\text{Legendre transform})$ is a good choice by asking for some specific property I want in the new formalism? Are there formalisms other than the Lagrangian and Hamiltonian ones that can be obtained via this sort of reasoning? (this part might be better asked in a different question if there is an answer that is independent of the context of this one, I'm not sure).
 A: In Lagrangian/Hamiltonian the crucial property of the Legendre transform is that it preserves all information. One way to demonstrate that the Legendre transform preserves all information is to show that it is its own inverse. When the Legrendre transform is applied twice you are back to the starting point.
The required properties of the transform are very specific, I rather expect that the Legendre tranform is the only transform that has the required properties.
It seems to me that in the end asking what makes the Legendre transformation especially useful is similar to asking what is special about integration by parts that makes it so useful for calculating integrals.

Recommended reading:
2007 article by R. K. P. Zia, Edward F. Redish, and Susan R. McKay
Making sense of the Legendre transform
Blog post by Jess Riedel:
Legendre transform
A: I second @garyp's advice of looking into thermodynamics, where the origin of Legendre transforms and their usefulness is clearer.
To answer your main question, It is not that Legendre transformations are useful, it is that momenta are in many situations more useful than velocities. And to switch to a momentum picture you need to do Legendre transformations.
Let's take the example of thermodynamic energy. Energy depends on entropy, the derivative of energy wrt entropy is the temperature. The temperature of the system is much easier to measure than its entropy, so you want to come up with an alternative function that will depend on temperature so that the entropy is its derivative wrt temperature. You realize that to do this you need to perform Legendre transformations and the new function you get is the Helmholtz free energy.
In mechanics reasoning is similar. Your Lagrangian depends on velocity, but momentum (the derivative of Lagrangian wrt velocity) is often a more useful quantity (e.g. in many cases it is conserved). So you want to find an alternative function that depends on momentum so that the velocity is its derivative wrt momentum.
