Don't confuse the breakpoint construction with the product structure of two Kripke structures or automata. In the breakpoint construction, there is a transition from a state A := {sa, sb, ...} to a state B := {si, sj, ...} with input x if you can reach each element of B from at least one state of A via input x. So not all states of A need to be able to react to input x! It suffices if there is one state that is able to react to x and in total all states of B need to be reached somehow by any state of A.
So to have a concrete example:
If you start at s2 and get the input b you can either go to s2 or to s3
=> reachable(s2, b) = {s2, s3}
If you start at s3 and get the input b there is no possible transition
=> reachable(s3, b) = {}
From that we get:
reachable({s2, s3}, b) = reachable(s2, b) U reachable(s3, b) = {s2, s3} U {} = {s2, s3}
This procedure is actually the same as with the subset construction, the only difference is the second state set in each qi that is about the final sets (of which I didn't talk about here).
To conclude I do the same for you second example, although it is a bit more complicated:
If you start at s0 and get the input !a you can either go to s0 or to s1
=> reachable(s0, !a) = {s0, s1}
If you start at s1 and get the input !a, the result depends on the value of b, so you either get the following:
=> reachable(s1, !a & !b) = {s1} or
=> reachable(s1, !a & b) = {}
However regardless of those values of b, the union of the reachable states will contain s1 in any case due to having s1 in reachable(s0, !a)
=> reachable({s0, s1}, !a) = {s0, s1}
