Special Relativity: Interpretation of the partial derivate of Stress-Energy Tensor

This question is based on Carroll’s book Spacetime and Geometry, specifically from page 33 to page 36.

In the upper mentioned section we define the Stress-Energy Tensor as:

The flux of the four momentum $p^mu$ across a surface of constant $x^nu$

Here lies the first problem: canonically the flux of a vector across a surface is a scalar, not a matrix or a tensor. So I don’t get this definition at all.

But nevertheless I understand that the Stress-Energy Tensor represent a generalization of the concept of mass and energy, so we can move forward for now. We then get the form of the Stress-Energy Tensor for a perfect fluid:
$$T^{munu}=(rho+p)U^mu U^nu+peta^{munu}$$
(where $rho$ is the energy density, $p$ si the pressure and $U$ is the four-velocity; keep in mind that we are working in flat spacetime). This is fine for me, however then we come to the following expression:
$$partial _mu T^{munu}=0$$
It is stated that the $nu=0$ component corresponds to the conservation of energy and the other three components correspond to the conservation of momentum; but no direct proof, or justification, of this statement is given. Using the definition of $T^{munu}$ previously cited: how can we show that the upper mentioned statement is true, or at least plausible?