Example of Cycle Notation and Elementary Transpositions.

Question

How does one achieve the yellow box from the the second line of this problem?

Answer 1

Note $s_1s_2s_1=(13)$.

Thus we have $$(56)(45)(34)(13)=(65)(54)(43)(31)=(65431)=(16543)$$

Answer 2

Conceptually it is probably simplest to start by computing the two-line representation (last part of the equality you display), by tracing each element from right to left through the product of transposition.

Once you have that, tracing the cycle structure of the permutation should be routine.