Intuition behind multiplying (or composing) permutations.

I'm trying to grasp the intuition for permutations and their multiplication. So far this has been my intuitive understanding: A permutation is merely a shuffling of the symbols. Take for example $\sigma ,\pi \in {S}_{4}$ given by, $\sigma =\left(\begin{array}{cccc}1& 2& 3& 4\\ 3& 2& 1& 4\end{array}\right)$ and $\pi =\left(\begin{array}{cccc}1& 2& 3& 4\\ 2& 4& 1& 3\end{array}\right)$

I could rewrite them as a 4-tuple: $\sigma =(3,2,1,4)$ and $\pi =(2,4,1,3)$ as permutations of {1,2,3,4} and so $\begin{array}{}\text{(\#)}& \pi \circ \sigma =(2,4,1,3)\circ (3,2,1,4)=(1,4,2,3)\end{array}$

I understand how to get the result. I know how to multiply (or compose) two permutations.

My Question: What happened in equation # and what's going on intuitively? What shuffled around when composition happened? What does the result of product mean with respect to $\pi $ and $\sigma $?

I'm trying to grasp the intuition for permutations and their multiplication. So far this has been my intuitive understanding: A permutation is merely a shuffling of the symbols. Take for example $\sigma ,\pi \in {S}_{4}$ given by, $\sigma =\left(\begin{array}{cccc}1& 2& 3& 4\\ 3& 2& 1& 4\end{array}\right)$ and $\pi =\left(\begin{array}{cccc}1& 2& 3& 4\\ 2& 4& 1& 3\end{array}\right)$

I could rewrite them as a 4-tuple: $\sigma =(3,2,1,4)$ and $\pi =(2,4,1,3)$ as permutations of {1,2,3,4} and so $\begin{array}{}\text{(\#)}& \pi \circ \sigma =(2,4,1,3)\circ (3,2,1,4)=(1,4,2,3)\end{array}$

I understand how to get the result. I know how to multiply (or compose) two permutations.

My Question: What happened in equation # and what's going on intuitively? What shuffled around when composition happened? What does the result of product mean with respect to $\pi $ and $\sigma $?