The concluding sentence sums up the main purpose of your essay reemphasizing your thesis (without repeating it word for word) and proving an overall sense of conclusion. Is this true?
This depends on which teacher (audience) you're writing for. Every writing teacher has different expectations and requirements for essays, I'm afraid, just like professional editors have unique expectations and requirements with professional writers.
I teach my students that the conclusion should sum up the main purpose of the essay without repeating it word for word, but that the final sentence or sentences should address the "So what?" factor. That is, it answers the question "Why should I [the audience] care or take action?" "Why does it matter?"
The final sentence certainly should provide an overall sense of conclusion, though. It should make the argument feel finished (instead of leaving the reader with the impression that you just stopped when you met your word count requirement). I cannot know for certain since I don't know your instructor, but I suspect he or she is going for "true" on this question.