If the LAST character of the string causes a transition to a FINAL state, then the string is recognized (accepted). Any thing else, and the string is not recognized (is rejected).
Here are the ways in which a string might be rejected:
As the string is processed, a final state might be reached several times. Final states can have transitions out of them. The automaton does not have to stop when it reaches a final state. What is required is for the last character of the string to lead to a final state.
Make the next state transition, as directed by the current character.
Is the input string recognized?