Hindman's theoremHindman's theorem : if I remember correctly, the original proof was elementary but very long and complicated; whereas a proof using an idempotent ultrafilter can be explained in less than a page.
Still with ultrafilters, there's also Tychonoff's theoremTychonoff's theorem : the original proofs, either Tychonoff's with complete accumulation points, or the one with the Alexander subbase theoremAlexander subbase theorem are somewhat technical and require some imagination.
The proof using ultrafilters is extremely straightforward and anyone who has learned about ultrafilters can find it with little to no imagination. It also has the advantage of showing that if you work with Hausdorff spaces, you don't need the full axiom of choice, whereas the Alexander subbase theorem uses Zorn's lemma indistinctly.
