I'm reading about quotient spaces from here:
also, there is attached a photo:
What I think I understood is I have to remove "duplicates" in order to obtain an injective function.
I can somehow accept this answer, but now I will take into consideration another example: $\mathbb{Z}_{5}$. I will write down also the definition of a quotient space: Let $X$ be a topological space and $\sim$ an equivalence relation on $X$. For every $x \in X$, denote $[x]$ its equivalence class. The quotient space of $X$ modulo $\sim$ is given by the set: $$X/\sim=\left\{[x]:x\in X\right\}.$$ Going back to $\mathbb{Z}_{5}$ we can say that $\mathbb{Z}/\sim=\mathbb{Z}_{5}=\left\{[0], [1], [2], [3], [4]\right\}$.
So, for the example from photo is more clear why we are doing this "division", but for $\mathbb{Z}_{5}$ example, it is not clear at all.. Is there an understanding method which is working for both exaples?
Think of the number line $\mathbb R$ as a helix over a circle $S^1 \subset \mathbb C$, where $\mathbb Z$ lies over $1 \in S^1$. In this sort of gluing visualization it is intuitive that $S^1$ arises as a quotient of $\mathbb R$ by the equivalence relation $x \sim y$ if and only if $x = y + n $ for some $n \in \mathbb Z$. Group theoretically: $$S^1 \cong \mathbb R / \mathbb Z.$$
Similarly, $\mathbb Z^5$ if we arrange $\mathbb Z$ in such a way that all multiples of $5$ lie above $1$, and more generally all numbers that are congruent to $k$ are above $e^{2\pi i/5}$. Projecting yields a surjective homomorphism $\mathbb Z$ onto a subset of $S^1$ isomorphic to $\mathbb Z_5$. Again, group theoretically: $$\mathbb Z_5 \cong \mathbb Z / 5 \mathbb Z.$$ This is, I would say, the gluing interpretation of the equivalence relation $n \sim m$ if and only if $n - m$ is a multiple of $5$.
Is this more or less what you were looking for or did I misunderstand you?