This may be more of a heuristic question but I feel like it should have some concrete answers. I want to understand why a pair of functors being adjoint is useful. How do we use that fact to prove things?
Suppose we have our two functors $F: C \rightarrow D$ and $G: D \rightarrow C$ with $F \dashv G$ with a bijection, $$ \Psi_{X,Y}: \hom_{D}(F(X), Y) \longrightarrow \hom_{C}(X, G(Y)) $$ natural in all the variables. We can obviously use this to get a morphism $\Psi(\phi): X \rightarrow G(Y)$ given $\phi$, but what meaningful things can we say about the map $\Psi(\phi)$ that make the notion useful for proving things.
To take a more explicit example, say $f:X \rightarrow Y$ is a morphism of nice schemes. We have an adjunction $f^* \dashv f_*$ between categories of sheaves of modules. So how is this used? Often in algebraic geometry we do have situations where we have a morphism into a direct image of sheaves and want to prove something about a morphism out of the inverse image. But I can never see how to say anything more than "such a morphism exists". Can anyone gives some concrete examples?
In my limited experience by far the most useful thing about adjoints is the fact that left adjoints preserve colimits and in particular that left adjoints are right exact.
One example where this is useful is having a quick argument for the fact that for a quasicoherent $\mathcal{O}_Y$-module $M$ $f^{*} M$ is a quasicoherent $\mathcal{O}_X$-module, as quasicoherent can be defined by having locally a rightexact sequence of the form $$\bigoplus_I \mathcal{O}_U \rightarrow \bigoplus_J \mathcal{O}_U \rightarrow M \vert_U \rightarrow 0$$
Similarly the fact that $-\otimes_A B$ sends a free $A$-module to a free $B$-module is a consequence of the tensor-hom adjunction.
I hope this explains a bit, why having adjoints can be useful. Sadly I don’t have an example, where the isomorphism $\psi$ is used explicitly. Sometimes you can translate commuting diagrams involving the left adjoint to commuting diagrams involving the right adjoint by using the naturality of the isomorphism $\psi$. However this is often a kind of Yoga and sadly I don’t have a specific example at hand...