Axler defines a quotient space as follows:
Suppose $U$ is a subspace of $V$. Then the quotient space $V/U$ if the set of all affine subsets of $V$ parallel to $U$. In other words, $$V/U = \{v + U : v \in V\}.$$
As an example of one, he gives:
If $U$ is a line in $\mathbb{R}^3$ containing the origin, then $\mathbb{R}^3/U$ is the set of all lines in $\mathbb{R}^3$ parallel to $U$.
Geometrically this should be a set of 'vertical lines' covering $\mathbb{R}^3$, but I do not know how to create such a plot nor do I understand where it came from. The best I have been able to do is work it out algebraically.
First, by a "line" in $\mathbb{R}^3$, I assume Axler means a subset of the form $$U = \{(x,0,0) \mid x \in \mathbb{R}\}.$$ Then we consider $\mathbb{R}^3/U$. Having fixed a point $(a,b,c)$, its equivalence class is the set of all $(x,y,z)$ satisfying: \begin{align*} (a,b,c) \sim (x,y,z) \iff (a - c, b - y, c - z) \in U \iff b - c = 0, c - z = 0 \iff b = c, c = z. \end{align*} It made more sense when I picked an explicit point and an explicit line, but the punchline should be that I get a line through $(a,b,c)$ parallel to $U$. But I do not know how to plot this or what "parallel" means in multiple dimensions.
You'd be better served with the "point-and-vector" description of a line (note that in $\mathbb{R}^3$, you would need two non-equivalent linear equations to determine a line; in $\mathbb{R}^n$ you would need $n-1$ linearly independent linear equations to determine a line).
A line in $\mathbb{R}^n$ can be described by a pair of $n$-tuples, a basepoint $\mathbf{b}_0$ and a "direction" $\mathbf{v}\neq\mathbf{0}$, $$\mathbf{b}_0 = (a_1,\ldots,a_n),\qquad \mathbf{v}=(v_1,\ldots,v_n).$$ The line consists precisely of all points of the form $$\mathbf{p} = \mathbf{b}_0 + t\mathbf{v},\quad t\in\mathbb{R}.$$ Geometrically, you are taking the vector $\mathbf{v}$, stretching/contracting/flipping it by a factor of $t$, and then placing its base at $\mathbf{b}_0$. The line consists of all the endpoints you can get this way. Algebraically, it consists of all $n$-tuples of the form $$(a_1+tv_1, a_2+tv_2,\ldots,a_n+tv_n),\qquad t\in\mathbb{R}.$$
Two lines, $L_1=\mathbf{b}_0+t\mathbf{v}$ and $L_2=\mathbf{c}_0+t\mathbf{w}$ are parallel if and only if the two vectors $\mathbf{v}$ and $\mathbf{w}$ are parallel, if and only if they are each a scalar multiple of the other.