To thicken is to become denser or to make something thicker. When you accidentally burn dinner, the smoke in the kitchen thickens. Time to get the fire extinguisher.
There are two ways to thicken: to become thick or to make thick. Cooks often use flour or cornstarch to thicken sauces. And when you add flour to your brownie batter, you thicken it. An earlier, now obsolete meaning was "to crowd together." The phrase "the plot thickens" means that a situation is becoming more and more complicated or interesting.