It is about the fact that the bisimulation relations on the same structure are equivalence relations and therefore reflexive and symmetric. Reflexive means that for every state s, there is a pair (s,s) included from the beginning and will remain there, so that it is nasty to always write it down. You may call the set of such pairs ID (for identity), and always write {...} ∪ ID without listing ID explicity.
Also, the pairs are symmetric which means that if there is (s,s'), then there is also (s',s). Again, you may skip half of the pairs if you note down that one should add the symmetric closure.
For example, the first step was
(S0,S0);(S0,S5);
(S1,S1);(S1,S2);(S1,S3);(S1,S4);
(S2,S1);(S2,S2);(S2,S3);(S2,S4);
(S3,S1);(S3,S2);(S3,S3);(S3,S4);
(S4,S1);(S4,S2);(S4,S3);(S4,S4);
(S5,S0);(S5,S5)
and becomes then
{(S0,S0);(S1,S1);(S2,S2);(S3,S3);(S4,S4);(S5,S5)}
∪
{(S0,S5);
(S1,S2);(S1,S3);(S1,S4);
(S2,S3);(S2,S4);
(S3,S4)}
∪
{(S2,S1);
(S3,S1);(S3,S2);
(S4,S1);(S4,S2);(S4,S3);
(S5,S0)}
which would be written just
ID
∪
{(S0,S5);
(S1,S2);(S1,S3);(S1,S4);
(S2,S3);(S2,S4);
(S3,S4)}^
where R^ should denote the symmetric closure.
