What is the purpose of the category of topological pairs?

A guiding principle in homotopy theory is that categorical notions such as limits and colimits of diagram are correct in spirit, but they fail to correctly capture homotopical notions.

So as a first approximation to why we should care about the category of pairs of spaces when we study homotopy theory of spaces, is that constructions such as $X/A$ naturally arise from the category of pairs of spaces. Of course, one can describe this construction without mentioning this category, but why avoid it when it comes up so naturally?

Now let us refine this reasoning. We ultimately care about the homotopy category of spaces, so we would like to make sense of what a homotopy invariant quotient should be. Our natural instincts should tell us that a pair $(X,A)$ should be equivalent to a pair $(Y,B)$ when we have a map $X \rightarrow Y$ that is a homotopy equivalence, and its restriction to $A$ sends us to $B$ via a homotopy equivalence (maybe there are subtleties about how the homotopy inverse should interact with the subspaces, let's ignore those).

The issue with the functor $(X,A) \rightarrow X/A$ is that it does not interact well with homotopy equivalences of pairs. For example, if one takes $X=S(\{1,1/2,1/3,\dots \} \cup \{0\})$, we may take the pair $(X,(\{0\},0))$ and include it into $(X, \{0\} \times I)$. This is a homotopy equivalence of pairs, but the quotients have different homotopy types.

This is where we can formulate more concretely this principle. If functors respect our "weak equivalences" (here the homotopy equivalences of pairs), then we have no issue defining homotopy invariant versions of these functors (just take it to be itself). However, it is often the case that functors do not respect our weak equivalences (as we just saw), but it is still important to have a reasonable definition. In the majority of cases we can proceed as follows: find some subcategory for which the functor does work well with and then show that we can functorially find a weak equivalence from any object in the category into/from a space in this subcategory. Then we define a homotopy invariant version of our original functor by making this functorial replacement and applying our original functor.

In this case, our functorial replacement will be $(X,A) \rightarrow (X, M(A))$ where $M(A)$ is obtained by gluing $A \times I$ onto $M$ via the $M \times \{0\}$. Then you will notice that the homotopy invariant version of $X/A$ is $X \cup C(A)$ (where $C(A)$ is the cone on $A$).

Now often we don't want to have to replace our object with a new one because our original object is the one we are interested in. In this situation, we need to argue that the functor applied to our original object is actually equivalent to the homotopically altered functor. Usually this will not be true, so we have to find special conditions.

In our case this comes down to asking when $(X, X \cup CA) \rightarrow (X/A,A/A)$, given by quotienting, is a homotopy equivalence. You will recall, that when excision applies this holds! So in fact, by introducing the category of pairs and their weak equivalences, not only have we motivated the definition of the cone on a subspace, but we have also motivated the comparison between the cone and the quotient, and we have motivated why it is important to know when this comparision is an equivalence.

This line of thinking is how we go about generalizing homotopy theory to situations far from topological, like chain complexes.

Two basic concepts of algebraic topology are homotopy groups and homology groups. In both cases it does not suffice to consider "absolute spaces" $X$.

In the case of homotopy groups we need a basepoint $x_0 \in X$ to introduce the fundamental group $\pi_1(X,x_0)$ and the higher homotopy groups $\pi_n(X,x_0)$ for $n > 1$. But okay, here we only consider special pairs of the form $(X,\{x_0\})$.

The standard approach to homology theory (as you can find it in Spanier's book and most other textbooks) is to define not only the homology groups $H_n(X)$ of spaces, but also the relative homology groups $H_n(X,A)$ of pairs $(X,A)$. Only this ingredient allows to formulate the exactness axiom and the excision axiom (see Spanier section 4.8). Without these homology axioms (or if you want, these properties of homology) you cannot calculate homology groups $H_n(X)$ of spaces, not even of simple spaces like the spheres $S^k$.

Singular homology satisfies these axioms. For excision see Spanier's Corollary 4.6.5, for exactness Theorem 4.5.4. Note that the latter is a theorem concerning chain complexes which implies the exactness axiom for singular homology. But surprisingly Spanier does not state this as an explicit theorem, he only mentions it in the text preceding Lemma 4.5.9 and once again after he axiomatically defined the concept of a homology theory.

I should not keep secret that there are alternative approaches to axiomatic homology theory which do not use relative homology groups. See for example Hatcher's "Algebraic Topology" section 2.3. But even in these approaches you need the category of pairs of spaces to formulate the axioms.

Conclusion: Algebraic topology without using pairs would be fairly unproductive.