TL;DR: It may be helpful to think of the above-diagonal orange components as "energy flux" rather than "momentum density"; if you do this, the interpretations in terms of shears and pressures become more natural.
Here's another way to think of the stress-energy tensor. First, you're hopefully familiar with the notion of the energy-momentum four-vector: $p^\mu = (E/c, p_x, p_y, p_z)$. Each one of the components of this quantity is conserved.
Second, you have hopefully come across some form of the continuity equation. This is a statement about conservation of some quantity that can flow through space. If this quantity has a density $\rho$ and a flux density $\vec{J}$, then we have
$$
\frac{\partial \rho}{\partial t} + \vec{\nabla} \cdot \vec{J} = 0.
$$
What this says, effectively, is that if the quantity [foo] is flowing out of a region of space (i.e., $\vec{\nabla} \cdot \vec{J} \neq 0$ at a particular point), then the density of [foo] must be changing at that point (and with the opposite sign). This is, for example, how charge conservation is enforced in classical electrodynamics; if we integrate the equation over some region and use the divergence theorem, we get
$$
\frac{dQ_\text{enc}}{dt} + \oint \vec{J} \cdot d\vec{a} = 0,
$$
i.e., if there's a net current flux through the surface, then the charge enclosed is changing.
The stress-energy tensor can be thought of as combining these two notions. If we want $E$ to be conserved, for example, and we allow energy to be spread out over space, then it must obey a law like
$$
\frac{\partial}{\partial t}\text{(energy density)} = - \vec{\nabla} \cdot \text{(energy flux)}
$$
and if we want momentum to be conserved, then each of the momentum components $x$, $y$, $z$ must also satisfy a similar law:
$$
\frac{\partial}{\partial t}\text{($p_x$ density)} = - \vec{\nabla} \cdot \text{($p_x$ flux)}
$$
If we look at the $T^{1 \mu}$ components of the stress-energy tensor, though, we have precisely these quantities! $T^{10}$ is the momentum density; and $T^{11}$, $T^{12}$, and $T^{13}$ are the fluxes of $x$-momentum in the $x$-, $y$-, and $z-$direction respectively. To see why this is, note that if $x$-momentum is fluxing in the $x$-direction across a surface, this means that the objects on the other side of the surface are experiencing a force in the $x$-direction (since $\vec{F} = d \vec{p}/dt$); and since momentum flux is just momentum per time per area, $T^{11}$ is just a pressure in the $x$-direction. By the same logic, if $x$-momentum is fluxing in the $y$-direction across a surface, this would correspond to a shear stress (a force is being exerted parallel to the surface rather than perpendicular to it.)
So that explains why the green components are pressures, the dark blue are shears, and the components $T^{10}$, $T^{20}$, and $T^{30}$ are momentum densities. But what about energy conservation? Well, if we try to do the same here, we can identify $T^{00}$ as the energy density; but under this interpretation $T^{01}$, $T^{02}$, and $T^{03}$ are more naturally thought of as energy flux rather than momentum density. It is apparently just a fact about the universe that these two quantities are equal to each other; at the very least, given the symmetries we know about between space, time, energy, and momentum, it should seem plausible that this is true.
Finally: all four of the above equations can be expressed pretty compactly as
$$
\frac{\partial T^{\mu 0}}{\partial t} + \partial_i T^{\mu i} = 0
$$
or even more compactly as
$$
\frac{\partial T^{\mu \nu}}{\partial x^\nu} = 0.
$$
Thus, we have a nice tensorial relation expressing the conservation of energy and momentum in our system.
