Finite state automata: final states
You seem to have a misunderstanding of generative models v.s. "recognizing" models.

The grammar you have on the right generates words by applying rules, starting from the initial variable, and stopping after there are no more variables.

Automata, however, recognize a language as follows: you feed the automaton a word, letter by letter, and the automaton takes transitions based on the letters given to it. 

If, after reading all the letters, the automaton ends up in an accepting (a.k.a final) state, then we say that the automaton accepts the word.

So it's better  to think of those as "accepting" states, rather than "final" states, although both terms are commonly used.