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 Gaussian correlation inequality, which is so simple that it has been a while for people to believe. Later people wrote another paper to explain it.

https://arxiv.org/abs/1512.08776