Here's how Fraleigh (informally) derives Fermat's Little Theorem in his classic Abstract Algebra text:
For $\mathbb{Z}_p$, the elements $$1, 2, \dots, p-1$$ form a group of order $p-1$ under multiplication modulo $p$. Since the $\color{blue}{\textrm{order of any element in a group divides the order of the group,}}$ we see that for $b \neq 0 \textrm{ and } b \in \mathbb{Z}_p$, we have $\color{red}{b^{p-1} = 1 \textrm{ in } \mathbb{Z}_p}$. Since $\mathbb{Z}_p$ is isomorphic to the ring of cosets of the form $a + p\mathbb{Z}$, we have $a^{p-1} \equiv 1 (\mod p)$.
Can someone please explain how the statement in red came about? Also, is the statement in blue coming from the Theorem of Lagrange?