Dvir's proof of the finite field Kakeya conjecture via the polynomial method was already mentioned in another answer. But I think the recent Croot-Lev-Pach/Ellenberg-Gijswijt resolution of the cap set problem via the polynomial method deserves to be mentioned as well. Before this breakthrough, extremely intricate arguments were needed to improve even slightly the upper bound on the size of the largest cap set (see e.g. the 2011 paper of Bateman-Katz referenced on the Wikipedia page). Whereas, the proof via the polynomial method is so simple and clean that any working mathematician can absorb it in an afternoon.