The most recent and astonishing one is Zeev Dvir's proof on Kakeya conjecture over finite field, it is surprisingly elementary and beautiful. It probably tells us sometimes it is just a change of thoughts we need to prove hard theorems.
https://terrytao.wordpress.com/2008/03/24/dvirs-proof-of-the-finite-field-kakeya-conjecture/
Another one is the elementary proof of the GCIGaussian correlation inequality, which is so simple that it has been a while for people to believe. Later people wrote another paper to explain it.