You’re right, I’m skipping over a few steps here.

What is happening is that we’re starting from L_{opt}, and moving the prefix of L that feerate-diagram-coincides with it to the front of L_{opt} so that we can skip over that, and just compare the part where the diagram actually differs. If it were any more obvious than it is, it would suffice to say “without loss of generality, assume L and L_{opt} have a distinct first chunk feerate”. This part of the argument is just to show that it indeed is without loss of generality, because if there is a coinciding prefix, then applying the theorem to just the transactions not in that prefix still leads to a contradiction.

This prefix p may consists of multiple chunks of L. If it consisted of 0 chunks, there is nothing to do, as L'_{opt} = L_{opt} in that case. If p consists of a single L-chunk, then just applying the gathering theorem gives our desired result (because in that case p has a feerate at least as much (and in fact, exactly equal to) the first chunk of L_{opt}).

The difficulty is only when p consists of multiple chunks. But in this case, repeatedly applying the gathering theorem (to move one L-chunk worth of p subset to the front of L_{opt}) and the stripping theorem (to skip over the just moved part) gets the result. I’m happy to write it out more formally, but this informal description is perhaps more enlightening?