The meaning of the word vindicate is <em>to prove that what someone said or</em> <em>did was right or true, after other people thought is was wrong </em>; the same as absolved; free from blame, etc. The fact that her brother admitted that he was responsible for the mess in the kitchen means that <em>she is clear, not guilty, and it is a proof that she was telling the truth.</em>
The word vindicated, which comes from the Latin word vindicatus, originally meant "to avenge or revenge" but its meaning soon shifted to "clear from censure or doubt, by means of demonstration." When you are vindicated, your name is cleared. You might also prove that you're right about something.
