How do I follow the text of this Longest Common Sequence proof?
The two red-underlined parts of the proof is to standardize an LCS solution, that is called $T_{opt}$ therein.

The idea is to push all the $0$'s farthest possible to the right (without passing through its nearest $1$ to the right).

Consider the following two strings:

$s_1=0001000100010001$ (i.e. $n=4$ vertices)

$s_2=0001000\ 00010001000$ (where the space is for visualizing purpose only) (i.e. $u=2$, $v=4$)

A common subsequence (not necessarily longest) can be: $s=0001000 00010001$ (i.e. $u$ is omitted)

The standardization process in the proof is to actually align $s$ against $s_1$, $s_2$.

And, this MUST be standardized to be as follows:

s1= 0001 0001 0001 0001

s2= 0001 000  0001 0001000

s = 0001 000  0001 0001

Similarly, if we omit $v$, then $s=000100010001000$

And, this should be standardized as follows:

s1= 0001      0001 0001 0001

s2= 0001 000  0001 0001 000

s = 0001      0001 0001 000

Finally, we will be able to construct an $n^2+k$-long common subsequence if there exists an independent set of size $n-k$, where for each edge $\{u,v\}$, we omit at least one of them (the ones that are in the independent set) so that it is possible to align like we do above (note that each edge-string $e_j$ contains only $n-1$ symbol $1$'s). And lastly, by complementing a $k$-VC, we have an $(n-k)$-IS.