It is not supposed to be a 2D space. It is supposed to illustrate a nested sequence of subspaces of the nullspace. All the subspaces $\mathcal M_4, \mathcal M_3, \mathcal M_2, \mathcal M_1, \mathcal M_0$ are subspaces of the nullspace of the matrix $L$ satisfying $\mathcal M_4 \subseteq \dots \subseteq M_0 = N(L)$. 4 is here is the largest exponent such that $L^k \neq 0$. Every dot in the figure is supposed to be a basis vector.
The text describes it by first finding a basis for $M^4$, then extending it to $M^3$, etc. until you reach $M^0 = N(L)$ and then have a basis for the whole nullspace.
The next picture (7.7.2) then describes how these vectors are extended to a whole basis for $\mathbb C^n$. If the vector $b$ is represented by a dot in $\mathcal M_i$ (but not $\mathcal M_{i+1}$), you can build a chain "on top" of this vector by solving $b = L^i x$ and then taking $L^{i-1}x, L^{i-2}, \dots, L x, x$ to be the chain formed by $b$. Thus, if $b$ is a basis vector for $\mathcal M_i$ (but not $\mathcal M_{i+1}$) it will have a tower of $i$ vectors on top of it (in fig 7.2.2).