What exactly is $T_{\mu\nu}$? Continuous matter is described in special relativity by the matter tensor which is the so-called stress-energy-momentum tensor. 
I am finding a difficulty understanding how a tensorial tool (mathematical tool) like $T_{\mu\nu}$ is the one that describes an idea so physical like the continuous matter. How can those two ideas relate?
 A: Perhaps there are two parts of your question.  First, why is the stress-energy tensor a useful and natural construct in relativity?  And, second, why are mathematical constructs useful and effective in describing physical systems.  The latter is largely a philosophical question that I think is too speculative to discuss here.*
Both special and general relativity are based on the principle of relativity which shows us that we can't think solely about something's velocity through space (i.e. 3D velocity $v_i$), but instead have to consider its velocity in space-time (i.e. 4D velocity $v_\mu$).  The 3D components of space blend with the dimension of time --- requiring them to be treated together.  For the same reasons, the traditional concepts of 'rest-mass' and spatial momentum cannot be considered separately --- and instead must be combined into the 'four-momentum'.
The four-momentum is a great description of a single particle, or single parcel of a fluid (for example).  But for continuum dynamics, we need to be able to include a description of how each of the components of the four-momentum move across each of the dimensions of space-time.  Just like @KyleKanos points out, this is exactly analogous to the (3D) stress-tensor in fluid dynamics --- allowing for the natural description of the 'flow' of something like x-momentum, in the y-direction.  This yields a tensor.
*But simply observing the effectiveness and ubiquity of mathematical descriptions should be sufficient to demonstrate their unarguable utility.
A: 
I am finding a difficulty understanding how a tensorial tool (mathematical tool) is the one that describes an idea so physical like the continuous matter. How can those two ideas relate?

That seems like asking how any mathematical tool can describe physics, which is unrelated to the stress-energy tensor itself or general relativity. The answer to such question is that mankind has invented a language to describe logical relations among things in the universe, which undergoes the name of mathematics (and all the flavours and significances related to it). After all, that is no more difficult to understand why displacements can be described by paths $ds$ or even more hardcore, why velocities are derivatives.
However, coming back to the stress-energy tensor itself, before going to general relativity it is worth noticing that it enters physics in many other ways already. In many engineering areas it describes stresses due to external energies and momenta applied to a system, relating them to the virtual and actual work that forces must do in order to even them out. Moreover, as already mentioned by Kyle Kanos above, fluid dynamics contains equations in terms of $T_{\mu\nu}$ essentially for the same reasons, only at the infinitesimal level.
In Lagrangian field theories the stress-energy tensor is defined taking derivatives of the Lagrangian with respect to the metric and it represent the translational contributions thereof in terms of energy and momentum (a more appropriate name of its is, in fact, energy-momentum tensor). Saying that the theory is invariant under translations (as every good physical theory must be, in particular general relativity since it is all about invariancies) means that $\nabla_{\nu}T^{\mu\nu}=0$. Accidentally, the Einstein tensor shows the same behaviour too, namely $\nabla_{\nu}\left(R^{\mu\nu}-\frac{1}{2}\,Rg^{\mu\nu}\right)=0$. An educated guess by Einstein was, then, that the two quantities might have been related, up to a scaling factor. So far this seems to be experimentally true, apart from refinements with cosmological constants or additional Lagrangian contributions to the action.
