In contexts like presenting a speech, writing a story, writing an ad, the word "twist" means to take something familiar and make it seem fresh and new by making a small change. The literal idea is that you aren't really changing something, you're just turning it around a bit. It's very similar to the phrase "to get a new angle on" something.
West Side Story took the story from Shakespeare's "Romeo and Juliet" and gave it a new twist by setting it in 1950's New York.
"Star Wars VII: The Force Awakens" is the same old classic Star Wars formula with a new twist.
Your speech should follow the pattern "Tell them what you're going to say, then say it, then tell them what you said." However, your conclusion shouldn't just restate what's been said, you should put a new twist on it.