Why the extra state in this automaton? Why does the following DFA have (to have) the state $b_4$?

Shouldn't states $b_1,b_2,b_3$ already cover "exactly two 1s"?
Wouldn't state $b_4$ mean "more than two 1s", even if it doesn't trigger an accept state?
$b_4$ is what is called a trap state, that is, a state that exists just so that all possible transitions are explicitly represented, even those that do not lead to a final state.

It doesn't change the language that is being defined, and can be omitted for the sake of brevity.