One of the most fundamental properties of tensors is that they are linear maps from vectors to vectors, for example $M:\vec x\rightarrow M\cdot \vec x$. There also tensors that are defined by the fact that they send two vectors to a scalar, for example the metric tensor: $g: (\vec x,\vec y)\rightarrow \vec x^T\cdot g\cdot \vec y$, but I will focus on linear maps from vector space to vector space for now.
A linear map can be identified by knowing how it acts on the basis elements of your vector space. Let $\color{red}{\vec e_1}=(1,0,0)^T,\quad\color{green}{\vec e_2}=(0,1,0)^T,\quad\color{blue}{\vec e_3}=(0,0,1)^T$. A linear map $M$ is then uniquely specified by the three vectors that are the result of applying $M$ to the basis vectors, i.e.
\begin{align}
\color{red}{\vec m_1}&=M\color{red}{\vec e_1}\\
\color{green}{\vec m_2}&=M\color{green}{\vec e_2}\\
\color{blue}{\vec m_3}&=M\color{blue}{\vec e_3}
\end{align}
In fact, these vectors can be easily found if we write this linear map in matrix form:
\begin{align}
M=\left(
\begin{array}{ccc}
\color{red}{M_{1,1}} & \color{green}{M_{1,2}} & \color{blue}{M_{1,3}} \\
\color{red}{M_{2,1}} & \color{green}{M_{2,2}} & \color{blue}{M_{2,3}} \\
\color{red}{M_{3,1}} & \color{green}{M_{3,2}} & \color{blue}{M_{3,3}} \\
\end{array}
\right)=\begin{pmatrix}\color{red}{\vec m_1}&\color{green}{\vec m_2}&\color{blue}{\vec m_3}\end{pmatrix}
\end{align}
We can plot three 3D vectors just fine. Can you tell which matrices I plotted in the picture below? An alternative to plotting 3 arrows is to plot an object like a cube subject to the transformation. A cube is very symmetric so when you do this you have to color code each face or otherwise use a less symmetric object to be able to distinguish reflections.
As a sidenote: for the moment of inertia tensor this picture might not tell too much. Maybe it would be nice if you could a show cube with the same moment of inertia as the object you are interested in.
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/3Ddxx.png)
I used the following Mathematica code to generate these:
plotmatrix[M_, pos_ : {0, 0, 0}] := {
Red, Arrow[{pos, pos + M[[1]]}],
Green, Arrow[{pos, pos + M[[2]]}],
Blue, Arrow[{pos, pos + M[[3]]}]
}
offset = 1.5 {-1, 0, 0};
Graphics3D[{
plotmatrix[T, offset*0],
plotmatrix[RotationMatrix[20 Degree, {1, 0, 0}], offset*1],
plotmatrix[DiagonalMatrix[{1, 2, .5}], offset*2],
plotmatrix[{{1, 0, 0}, {0, 1, 0}, {0, 1, 1}}, offset*3]
},
AxesLabel -> {x, y, z}, Axes -> True]