I'm also looking forward to any answers to this question, but no one answered so far. So I'm trying to provide some viewpoints.
The functor $\pi_0$
Let's look at a simpler example: the functor $\pi_0:\mathbf{Top}\to \mathbf{Set}$ defined in Chapter 2.1.
The functor $\pi_0$ sends a topological space $X$ to the set of its path components, and a continuous function to the induced set map. If we consider New Zealand as a topological space $X$, then $\pi_0(X)$ would become a two-element set $\{\textit{North Island, South Island}\}$ (ignoring other smaller islands).
After applying $\pi_0$, every pair of points in $X$ that is connected by a path are shrunk to a same element in $\pi_0(X)$:
Since all paths are glued onto two points (precisely: elements), the notion of path has vanished in the RHS above and discrete information of the original space (the number of path components) has shown.
In $\mathbf{Top}$ we first relate points/mappings/paths by a topological concept, then glue the related objects together, and finally get a new (often discrete) structure where the topological concept we initially introduced vanishes.
It is just like: in algebra we quotient a group $G$ along its commutator subgroup $C_G$, and get an Abelian group $G/C_G$, in which all commutators vanish (become identity).
The functor $\pi_1$
In advance, let the annulus in the following picture be a pointed topological space $X$. The fundamental group functor $\pi_1:\mathbf{Top^0}\to\mathbf{Grp}$ makes every class of paths that is related by homotopies in $X$ become a single element in $\pi_1(X)$:
The paths which are related by a topological concept (homotopy) are shrunk to be a single element in $\pi_1(X)$. Therefore, roughly speaking, in $\pi_1(X)$ no two distinct elements are related by a homotopy now (a topological concept vanishes).
Eliminating such topological concept in a space yields a discrete structure $\pi_1(X)\cong (\mathbb{Z}, +)$. Operations defined for classes of paths (eg. concatenation) may look like manipulating discrete elements in $\pi_1(X)$.
From $\mathbf{Top}$ to $\textbf{h-Top}$
Likewise, in the category $\mathbf{Top}$, there are continuous maps that can be associated by homotopies (in some sense, associated by structures of continuum):
In $\textbf{h-}\mathbf{Top}$, these maps are quotiented together:
Thus, (maybe more than) a continuum of morphisms are packed into one. We take a "bunch" of morphisms at once, and study interactions between "bunches". These bunches again considered to be discrete, since no pairs of "big" morphisms are homotopic anymore.
