For a group $G$ of order $p^n$, $G\cong H$ for some $H\le\Bbb Z_p\wr\dots\wr\Bbb Z_p$.
By Cayley's Theorem, we can embed $G$ into the symmetric group $S_{|G|} = S_{p^n}$. So by Sylow's Theorem, $G$ is isomorphic to a subgroup of a Sylow $p$-subgroup $P$ of $S_{p^n}$.
(I prefer $C_p$ to ${\mathbb Z}_p$ for a cyclic group of order $p$.) It is a standard result that $P \cong C_p \wr C_p \wr \cdots \wr C_p$ (with $n$ wreath factors.) Let's write that as $C_p^{\wr^n}$.
That is not hard to prove by induction. Assuming that $S_{p^{n-1}}$ has Sylow $p$-subgroup $Q \cong C_p^{\wr^{n-1}}$, by partitioning the set $\{1,2,\ldots,p^n\}$ into $p$ sets of size $p^{n-1}$, we see that $S_{p^n}$ contains the permutation wreath product $Q \wr C_p \cong C_p^{\wr^n}$ and by computing its order you can check that it is a Sylow $p$-subgroup.