Variational Principle What does the following statement mean, that Nature likes to minimize things (like energy) and this equation describes one particular minimization problem? 
What is the Variational Principle?
 A: The variational principle is, in fact, a physical principle. To get an idea, the easiest example of a variational principle is Fermat's principle in non-relativistic optics (the one taught at school via geometric means). 
The idea to use a variational principle is that at the end we get an equation, or even a set of equations, that will define the dynamics of the system for all times.
I am not sure if I understand your question, but if you mean to ask 'how do we come up with variational principles', I would guess by experiments. By experiments, we can notice that light tends to travel in a straight line (in non-relativistic experiments, that is!), so then, with this idea we can derive some equations. The same happens for the principle of least action, first you build some intuition about the behaviour of objects in dynamics, and then you find some equations which satisfy the physical model using those constraints.
If you check Landau-Lifshitz's book for classical mechanics (a fantastic read, by the way), you will see that the authors just go straight ahead and integrate the action, thus using a variational principle. Then they get the Euler-Lagrange equations.
A: It is, to some extent, just folklore. In particular, Nature likes to find stationary points, which can be maxima or saddle points, not only minima.
Usually we first meet variational principles when studying mechanics (maxima/minima of the action functional), and it is proved that most laws of general physics are variational in nature. This means, roughly, that solutions to the equations of motion are stationary points of some functional (like the action or the energy of a system).
The mathematical framework is considerably complicated: you need (infinite-dimensional) normed vector spaces, Fréchet and Gateaux derivatives, integration theory, some theory of ODEs and PDEs. The keywords are calculus of variations and critical point theory, but a good background is needed to approach these topics.
The Wikipedia page about variational principles is an interesting starting point.
