The Muppets employ various visual gags that puppets might not otherwise seem able to easily achieve. As there is most often only a hand inside the puppet working the mouth, blowing air or drinking liquid requires some trickery hidden from the camera.
The gags are not too difficult to figure out, but on-screen commentary from Kermit in The Muppet Show episode 307 seems to indicate a particular joy held by Jim Henson and his team in deploying them. After Kermit blows out a candle, he looks to the camera and says, "Think about that, folks."
Other gags are more practical. In Cookie Monster's case, the nature of his character requires him eat any number of items on-camera. While his consumption of cookies often includes a large portion of the treat flying almost anywhere but in his mouth, when more solid objects are eaten, a hole in the puppet's mouth accommodates the illusion of devouring the item.