$\quad$It is not difficult to show how category theory works on algebraic geometry by giving some examples, such as “the category of affine varieties is equivalent to the category of finitely generated k-algebras” somehow explains why algebraic geometry is based on commutative algebra; “maxSpec is not functorial, but Spec is” implies the motivation for the definition affine $n$-space and so on.
$\quad$Besides, I think there is another perfect way to make this problem understood——to study about Grothendieck.
$\quad$Why Grothendieck’s theory succeeded? He himself attributed some of the reasons to the development of category theory. In fact, Serre’s discussion on coherent sheaf in FAC was very successful at first, but he mainly used the tools of etale space. This leads to some complicated and unclear proofs. However, Grothendieck in his EGA made clear that he won’t use etale space, but the language of category theory. This move make it possible to just talk about presheaf without etale space and sheaf, meanwhile contributing to some new discoveries in sheaf theory.
$\quad$And all this is for solving Weil Conjecture, Serre told Grothendieck that Weil Conjecture can be solved by establishing a cohomology theory on Weil’s “space”, in which Lefschetz Fixed Point holds. But the properties of Weil’s space is too bad to build a cohomology theory because it doesn’t has separation property.
$\quad$After a period of time, people came to understand the essence of sheaf cohomology and Cech cohomology by the tools from homological algebra. Then something like Grothendieck topology, even topos theory was invented by Grothendieck. And all this work are closely related to homology, whose foundation is totally category theory.