The definition of a tensor of rank n in a three dimensional space is an array of $3^n$ values.
Therefore a tensor of rank $n=1$ in 3D space is a vector (made of three values, each corresponding to a value along axis defined by basis vectors), making a sample vector:
$$V = (1,2,3)$$
possible to interpret graphically as:
So far, so good. Now if I have a sample tensor of rank $n=2$ defined in a 3D space:
$$T=\begin{pmatrix} 1 & 2 & 1 \\ 3 & 4 & 0 \\ 5 & 3 & 2 \\ \end{pmatrix}$$
What would be its interpretation on 3D graph?
The vector space of rank $k$ tensors on an $n$-dimensional real vector space $V$ has dimension $n^k$, so you could visualize a rank $k$ tensor on $V$ as an arrow in $\mathbb{R}^{n^k}$.
However, thinking of a tensor (or what physicists mean by tensor) as an array of numbers isn't a very enlightening perspective. Sure, you can define a tensor by specifying its components, but this perspective doesn't help you understand what it's used for. Think of a rank $k$ tensor $\phi$ on a real vector space $V$ as a multilinear map $\phi: V \times V \times \cdots \times V \to \mathbb{R}$. That is, a tensor of rank $k$ takes $k$ vectors from $V$ as inputs, and is separately linear in each slot.