Why do black holes need a quantum mechanical description? I read about black holes, about the Schwarzschild metric, Einstein field equations and their solution in the vacuum for a spherical body.
I understood black holes are object whose gravity is enormous, and I also understood that there does exist the so called Schwarzschild radius, which is 
$$R_s = \frac{2GM}{c^2}$$
which is the radius of a sphere such that, if all the mass of an object were to be compressed within that sphere, the escape velocity from the surface of the sphere would equal the speed of light.
So I understood of course why General Relativity is more than essential to describe those objects.
But now, if we make an example with our Sun (which won't ever become a black hole, but let's just play), we would get
$$R_s \approx 2.95\cdot 10^3\ m$$
Now this sphere seems everything but infinitesimal to me, so my question is: why black holes do need also quantum mechanics to be completely understood? Or is it referred only to the "inside" of a Black Hole (namely the region beyond the Events Horizon)?
I'm sorry, this question may be unclear or stupid but don't down vote it. I'm sure you all got some doubts in your past, and the most beautiful thing is someone who explains them to you. 
 A: There are lots of reasons why we expect that a quantum theory of gravity is necessary, and they are well summarised in the question that ACuriousMind linked: What are the reasons to expect that gravity should be quantized?.
Bit since you are specifically asking about black holes let's just consider black holes, in which case the obvious reason we need a theory of quantum gravity is to describe Hawking radiation and black hole evaporation.
Hawking based his calculation on a technique called semiclassical gravity which is a sort of halfway house between classical physics and a full theory of quantum gravity. Semi-classical gravity is only an approximation, and while we expect it to work pretty well for large black holes it will get increasingly inaccurate as the block hole gets smaller. The only way we will ever understand what happens in the final stages of black hole evaporation is if we have a full theory of quantum gravity.
There's another aspect of black hole event horizons that is currently rather controversial and that's the black hole firewall paradox. If the firewall idea is true then the inside of a black hole is a weirder place than anyone suspected, but again with a proper theory of quantum gravity we'll never know.
These are a couple of examples of why we need a quantum gravity theory that are specific to black holes and which we may one day be able to actually measure. But I encourage you to read the question ACuriousMind linked for a deeper understand of why a classical theory of gravity and a quantum theory of matter are never going to sit well together.
A: You ask 

Why do black holes need a quantum mechanical description?

But do not give a link connected to this question. Quantum mechanics has been necessary for the current standard cosmological model, the Big Bang model.

There quantization of gravity is necessary for the model to fit the observations, but as far as I know, the general relativity version of black hole formation and description is adequate. Quantization of gravity is the holy grail of current theoretical trends in order to be able to unify all forces at the high energies of the beginning of the Big Bang. It is actively researched. At present only effective quantization models are used . String theories have  quantized gravity but there are many possibilities; no  unique theoretical model has been proposed up to now, that could contain the standard model of particle physics and give testable predictions.
