We can decompose any element of $S_p$ into the form $(a_1 \ \ b_1)(a_2 \ \ b_2) \dots (a_i \ \ b_i)$. If for some $1 \le j \le i$, $|a_j - b_j|=1$, it's easy! Because: Let $\sigma = (12 \ldots p)$. Then $\sigma^k (12) \sigma^{-k} = (k+1 \ \ k+2)$. But I can't write $(a_j \ \ b_j)$ as arbitrary products of $(12 \ldots p)$ and $(12)$ if $|a_j - b_j|>1$.
Any answer that I read in MSE was not easy to understand. I would appreciate any simple clear detailed explanation.