The way I think about quotient spaces (or quotient algebraic structures in general) is as an identification of things which differ by some subspace/subgroup/subring/... of the original structure. Then the quotient space (resp. algebraic structure) can be thought of as what you get when you squash the subspace (resp. substructure) to a point and extend to the rest of the space (resp. structure).

To illustrate this, with $\mathbb{Z}/n\mathbb{Z}$, we identify integers which differ by a multiple of $n$. What we do is squash $n\mathbb{Z}$ to a point, namely $0$, and this extends to the rest of the space by squashing $a+n\mathbb{Z}$ to $a$ for $0 \le a < n$. Then the operations of addition and so on are inherited naturally from the quotient operation.

As another example, with $\mathbb{R}^2/\langle (1,1) \rangle$ we identify things which lie on the same line lying at $45^{\circ}$ to the (positive) horizontal, i.e. we identify vectors which differ by some scalar multiple of the vector $(1,1)$. We can visualise this as contracting $\langle (1,1) \rangle$ to a point, which when you extend linearly contracts $\mathbb{R}^2$ to a line through the origin, leaving you with $\langle (1,-1) \rangle$. (Imagine squashing the whole plane down towards the origin perpendicular to the line $\langle (1,1) \rangle$). Another perspective is that the 'points' in $\mathbb{R}^2/\langle (1,1) \rangle$ can be thought of as precisely the lines in $\mathbb{R}^2$ with gradient $1$.

More generally, if $V$ is a vector space and $U \le V$ is a subspace then you can think of $V/U$ as the space you get when you identify two elements $v, v' \in V$ if $v'=v+u$ for some $u \in U$. What it 'looks like' is what you get when you contract $U$ to a point and extend linearly to the rest of the space.

I hope this wasn't too waffly.