Singular homology can be motivated by the desire to define invariants of topological spaces. Intuitively it measures the holes in a topological space.
Question: What's the most natural way of motivating singular cohomology? Okay, it is an invariant too. But what does it measure?
I think "how to motivate singular cohomology" and "what does singular cohomology measure" are two pretty different questions. There is not really a geometric explanation for what singular cohomology measures as straightforward as for what singular homology measures. There is a nice explanation at the start of chapter 3 in Hatcher's text, which motivates singular cohomology as measuring failure of a certain "discretized differential equation" to be solvable, but, truthfully, this is not usually how one works with singular cohomology in practice. One could also make various points about how singular cohomology is related to other cohomologies such as de Rham cohomology, sheaf cohomology or Čech cohomology and then motivate those, but I feel like that's venturing too far. In my opinion, singular cohomology is simply not as intuitively accessible as singular homology. This is a hurdle one has to overcome.
The purely algebraic reason why singular cohomology is introduced is that it's a much better invariant. Namely, the cohomology groups can be summed to yield a graded object $H^{\ast}(X)=\bigoplus_{n\ge0}H^n(X)$, which turns out to support a natural graded algebra structure, the cup product. [Advanced Comment: this algebra structure actually comes from a coalgebra structure on the singular chain complexes, which however does not induce a coalgebra structure on the graded homology groups, as discussed here.] More algebraic structure means this is a stronger invariant. Indeed, the classic example is that $\mathbb{CP}^2$ and $S^2\vee S^4$ have isomorphic homology and cohomology groups, but you can detect that these spaces are not homotopy-equivalent by showing their cohomology rings are not isomorphic. Generalizing this leads to applications of cohomology to studying the homotopy groups of spheres. If you're studying manifolds, Poincaré duality presents a non-trivial relationship between cohomology and cohomology, which can be interpreted geometrically in multiple ways. In certain situations, Poincaré duality allows us to interpret the cup product on cohomology as "dual" to the intersection of submanifolds, which makes it relevant to intersection theory. The ring structure on $H^{\ast}(\mathbb{CP}^n)$, for example, has a concrete geometric interpretation in this sense. Even more generally, the reason cohomology behaves better than homology is that it is a representable functor in a well-understood way (homology is not representable on the other hand). This makes natural transformations between these functors accessible, which leads to cohomology operations, and hence even more structure.
There are also geometric reasons for introducing singular cohomology. Namely, there exist scenarios in nature, where geometric information is naturally encoded cohomologically, rather than homologically. For example, let $X$ be a path-connected (this is just for the sake of convenience) topological space and $\pi\colon E\rightarrow X$ a vector bundle. You can form the frame bundle $F(E)\rightarrow X$, which is a $\mathrm{GL}(n)$-principal bundle. Taking the quotient by the induced action of $\mathrm{GL}^+(n)$ yields a $\mathbb{Z}/2\mathbb{Z}$-principal bundle $O(E)=F(E)/\mathrm{GL}^+(n)\rightarrow X$, which is the same thing as a $2$-fold covering of $X$ with a distinguished deck transformation. The fiber over $x\in X$ consists of both vector space orientations on the vector space $E_x=\pi^{-1}(x)$ and the deck transformation switches all these orientations. The vector bundle $\pi$ is orientable if and only if $O(E)$ has a global section if and only if $O(E)$ is trivial if and only if every loop in $X$ lifts to a loop in $O(E)$.
From this, we obtain a $1$-cochain $C_1(X;\mathbb{Z})\rightarrow\mathbb{Z}/2\mathbb{Z}$, which takes a $1$-chain, writes it as a sum of loops and non-closed paths in a maximal way, maps the non-closed paths to $0$ and a loop to $0$ if and only if it lifts to a loop in $O(E)$. This vanishes on boundaries, because the $2$-simplex $\Delta^2$ is contractible, hence lifts to $O(E)$, so it is a $1$-cocycle, which defines a cohomology class $w_1(E)\in H^1(X;\mathbb{Z}/2\mathbb{Z})$. This cohomology class vanishes if and only if the $1$-cochain vanishes on cycles if and only if every loop in $X$ lifts to $O(E)$ if and only if $E$ is orientable. The geometric property of orientability is naturally encoded as a cohomology class. If $f\colon Y\rightarrow X$ is any continuous map, you can check that $f^{\ast}w_1(E)=w_1(f^{\ast}E)$, so these classes are well-behaved. Generalizing these ideas leads to the subjects of characteristic classes and obstruction theory (my presentation above is not the same thing as what is done in general, but I thought it more convenient to illustrate my point). You can check out chapter 3 of Hatcher's other text for some of the general picture.
