Proof: By the primality of $p$ and the Sylow theorems, we have that the $p$-Sylow subgroups are all generated by elements of order $p$ that are conjugate to each other.
Therefore, we can take without loss of generality our element of order $p$ to be the cycle $(234\dots p1)$. Let our transposition be $(ij)$. We can conjugate by our cycle to generate any transposition of $k$ and $k + j - i$. Then we can compose these transpositions of size $j - i$ starting at $1$, using primality at $p$ to get a transposition between $1$ and $h$ for $h < j - i$. This continues until we get the transposition $(12)$. Conjugating by our cycle repeatedly gives us all of the transpositions between $i$ and $i+1$, which then generate $S_p$.