I can certainly try!
My understanding of first-order logic is that it’s intended as a sort of purely logical language philosophers and other logicians can use to puzzle out logical arguments and proofs. I say “proofs” because FOL is, by its nature, nearly mathematical; most of my work in the class involved demonstrating “if A then B” but in long, long step-by-step proofs with plenty of subproofs. So a rough schematic of the sort of FOL I usually saw was, “If A then B, C, or D. If A then not C. If A then not D. If A then B.”
It’s not a coincidence, I think, that this mirrors the style of a lot of philosophical arguments. And I know that students working on the graduate level use the language to make arguments over things like the nature of determinism (David Foster Wallace did this in his senior thesis). But I think its greatest interest to many philosophers involves using it to settle arguments over the nature of logic itself, or as a battleground for disputes over how logic works.