I read about Willmore Energy is a quantitative measure of how much a given surface deviates from a round sphere. Also, I heard that it says that things in nature tends to change their shape in such a way that they use the least energy to survive. And it ends up to be a sphere by using Willmore energy. But how does this link to this formula?
$$W=\int_SH^2dA-\int_SKdA$$
I mean, what does this has to do with all of the real-life interpretations?
The formula encodes exactly the intuitive principle that a surface resists bending or stretching, in the simplest possible way.
H is the mean curvature and K is the Gaussian curvature. There should in principle also be something like a bending rigidity coefficient multiplying each term, to give the correct units and to weight the energy cost of each type of curvature by how much the surface resists each type of deformation. The terms are as you say something like two types of elastic bending energy, but instead of resisting compression or shear we have two terms resisting roughly "bending" (mean curvature) and "stretching" (Gaussian curvature) of a surface.
Look at examples of what surfaces with different mean and Gaussian curvatures look like. Roughly the first term means that the object resists the kind of sideways bending which would not distort a grid drawn on the surface in either directions, such as rolling a plane into a cylinder. The second term means that it resists bulging outward or inwards in a way which would distort a grid drawn on the surface. (Paper happens to be a material which can bend but does not stretch: anything you can do to a piece of paper is the first type of bending, which gives it mean curvature and not Gaussian curvature).
If we are only talking about spheres and deformations thereof, we can ignore the K integral term because the Gauss-Bonnet formula says that, while the topological genus doesn't change, the total Gaussian curvature over the surface, as obtained by integrating over the entire surface area, remains constant.
The lowest energy state is then the one where the first term is at a minimum, and the lowest value it can obtain is zero.
A sphere is a minimal surface with mean curvature (the first term) equal to 0. There are other minimal surfaces, and it is well known that objects such as soap films will form some minimal surface.
edit: more on mean curvature term
The mean curvature is half the sum of principle curvatures, $H=\frac{1}{2} (c_1+ c_2)$. To get the values you have to calculate the shape operator in some coordinate system and its eigenvalues; the most intuitive way to explain it is a sideways bending, like the difference between a flat surface and the surface of cylinder.
Since the mean curvature is squared we have $H^2 = \frac{1}{2}(c_1^2 + 2c_1c_2 + c_2 ^2)$
The middle term is the Gaussian curvature $K= c_1 c_2$; for a constant-genus surface we can also ignore the integral over this part. In the form given with $-\int K dS $ it is also conveniently subtracted off, leaving $W = \int (c_1^2 + c_2^2 ) dS$.
