A definition of the existential and universal predecessor states is as follows:
pre∃(Q2) := {s1 ∈ S | suc(s1) ∩ Q2 != {}}
pre∀(Q2) := {s1 ∈ S | suc(s1) ⊆ Q2}
and you know that the semantics of <> is pre∃ and that of [] is pre∀.
Hence, s1 must be in pre∃({s0,s2,s3,s4,s5,s6,s7}) since it must contain all those states s1 where we have suc(s1) ∩ Q2 != {}, i.e., that have at least one successor in Q2={s0,s2,s3,s4,s5,s6,s7}. Obviously, we have suc(s1) = {s5}∈Q2={s0,s2,s3,s4,s5,s6,s7}, and thus s1∈pre∃(Q2).
For pre∀({s0,s2,s3,s4,s5,s6,s7}) we should compute those states where all successor states are in {s0,s2,s3,s4,s5,s6,s7}, i.e., pre∀({s0,s2,s3,s4,s5,s6,s7}) = {s0,s1,s3,s4,s5,s6,s7}.