Group each nonzero element with its inverse. These become 1 and go away. This is fine except for values of x that are their own inverses. In other words, x2 = 1. We know there are only two square roots, 1 and -1, so after inverses pair off and go away, -1 is left. This is a special case of a more general theorem in group theory.
Another proof uses a primitive root r. since order doesn't matter, we are multiplying r1 × r2 × … × rp-1. The exponent is the sum of the integers from 1 to p-1, which is p×(p-1)/2. We know r to the (p-1)/2 is -1, and raising that to the p still gives -1.