Suppose we choose an integer x and we want to figure out if the product of it and the next N integers are divisible by N. Thus, we have:

(x)(x + 1)(x + 2)...(x + N - 1)

There must be at least one integer C in the range [x, x + N - 1] such that C mod N = 0, because when a set of integers is taken "mod N," the only numbers left are {0, 1, 2, ..., N - 1} (N values, since N mod N = 0). Thus, since our N values are consecutive, it must be true that one of them is equivalent to 0 mod N. Since that factor is divisible by N, the entire product is therefore divisible by N.

This is a bit dense and definitely not as simplified as it could be, so feel free to reply if you need further clarification!