Saying you're going to slide the dirt out of your mouth is just another way of saying you're going to announce something. All that information is the dirt, and to announce that dirt you just slide it on out of your mouth. How's that for a visual?
Now that you're about to announce something important, you should maybe get it nice and smooth in your mouth so the dirt can slide out easier. So take a big gulp of Coke (こく). Coke is just like, the dirt of soda anyway, you know?