The fundamental theorem of covering spaces states that for a nice topological space $X$, there is an equivalence of categories between covering spaces over $X$ and left $\pi_1(X)$-sets. "Grothendieck's Galois theory" states that is is even true for $X$ a connected scheme, if we replace $\pi_1(X)$ be its étale analogue, covering spaces by finite étale covers, and left $\pi_1(X)$-sets by finite continuous left $\pi_1(X)$-sets.
Certainly these are nice statements, because they both show that two a priori different stories ($G$-sets vs. coverings) turn out to be the same. However, I keep wondering:
QUESTION: What is the actual motivation people (could) came up with these statements? This could be answered by: What are some applications of these theorems?
A more down-to-earth formulation used in algebraic topology courses is the "local" version using posets instead of categories (isomorphism classes of covering spaces $\cong$ conjugacy classes of subgroups of $\pi_1(X)$), as discussed in Hatcher's book Theorem 1.38. I skimmed through this section of Hatcher's book but I can't find any concrete motivation except "here's a nice theorem / classification result: ...".