Intuition in permutations for Laplace Determinant Expansion

Question

Starting with the Leibniz formula for the determinant, I wish to derive the Laplace (Cofactor) Expansion. At the risk of being overly verbose, please see the proof here. Now I understand the idea of the proof. The difficulty I have is in how they managed to obtain that $$\tau\,=\left(n,\ n-1,\ \ldots,\ i\right)\circ\sigma^\prime\circ\left(j,\ j+1,\ \ldots,\ n\right)$$ I can see how $\sigma$ (and hence $\sigma^\prime$) is obtained from $\tau$ by shifting the indexes. However, I am unable to make an intuitive derivation of the above formula. If anyone can help shed some light that would be greatly appreciated.