The original proof used Cauchy-Davenport lemma. Several proofs are given in this article of Alon-Dubiner (The proofs deal only with the case when $n$ is prime, but deducing the general case is straightforward from there). Note that the ideas behind most of these proofs could be interpreted as special cases of the more powerful theorem that is commonly known as "Combinatorial Nullstellensatz" (proven by N. Alon, see here). The keyword for results like these is "Zero-sum Ramsey theory".
ETA: You might also find the paper by Olson, "A combinatorial problem in finite abelian groups", Journal of Number Theory (1969) Vol.1 very interesting. It proves a generalization of EGZ theorem for finite abelian p-groups (I think this was one of the first among many other generalizations).