Cyclicity of unit group of $\mathbb{Z}/p\mathbb{Z}$ for odd prime $p$ using $p$-adic exponential function

Question

I read this MSE answer on why the unit group of $\mathbb{Z}/p^n\mathbb{Z}$ for odd prime $p$ is cyclic. I do understand that $\exp_p$ is an isomorphism from $p\mathbb{Z}_p$ (additive) to $1 + p\mathbb{Z}_p$ (multiplicative). However, I do not understand the exact steps paul garret did (how do they conclude that the quotient must be cyclic? what does index-$p$ part mean? what is a (pro-) generator?). I would greatly appreciate it if someone could provide a more elaborate and detailed answer, or provide some more (different) $p$-adic proofs.