More precisely, a finite $p$-group $G$ of order $n$ embeds into $GL=GL_n({\mathbb F}_p)$ (Cayley theorem). The subgroup $UT$ of upper triangular unipotent matrices of $GL$ is a Sylow p-subgroup of $GL$ (proof by computing the order of $GL$), so $G$ is conjugate to a subgroup of $UT$.
Yes, of course, it is how Bogopolsky proves Sylow theorems in his book (see MathSciNet). More precisely, a finite $p$-group $G$ of order $n$ embeds into $GL=GL_n({\mathbb F}_p)$ (Cayley theorem). The subgroup $UT$ of upper triangular unipotent matrices of $GL$ is a Sylow p-subgroup of $GL$ (proof by computing the order of $GL$), so $G$ is conjugate to a subgroup of $UT$.