Reducing $x^{p-1}$ mod $p$

Question

I read a comment in an algebraic number theory text that the reduction of $f=x^{p-1}-1\in\mathbb{Z}_p$ reduces in $\mathbb{F}_p[x]$ to a product of distinct linear factors. Why is this true? I know that $x^{p^n}-x$ is the product of all the distinct irreducible factors in $\mathbb{F}_p$ of degree $d$ where $d$ runs through all divisors of $n$, but not sure if and how this applies.