b just means that b holds at the current point of time, and we don't have any statement about b's value at any other point of time.
F b means that b holds at some time in the future, that may be at time t=0, but also any other point of time.
In the above structure, EG!a holds in states s1 and s0, and since these states do not satisfy b, there is no state satisfying b&EG!a, and therefore also EF(b&EG!a) is nowhere satisfied.
However, S1 is satisfied in state s0, since there is the path s0 s2^omega, that satisfies Fb, and the first state of that path, i.e., s0 satisfies EG!a.
