I find the style of the proof a bit colloquial, it should be cleared up.
An alternative proof is the following:
Let S be the set {1,2,.., p-1} multiplication with a mod p induces a permution on S. Hence their products are equal,
1*2*..*(p-1)= (a*1)*(a*2)*...*(a*(p-1)) mod p
Since p is prime, the product 1*2*..*p-1 can be divided out and the result follows.
While we're at it we might as well refer to Lagrange's theorem and be done with it.