It's a common phrase used to show the importance of punctuation. It's from a joke about a panda bear (there are other threads here with the joke) and then it was the title of a book about punctuation.
In its non joke form: A panda bear eats [bamboo] shoots and leaves. A shoot is a new plant.
In the joke, a panda orders food at a restaurant, and after eating, get up without paying, fires a gun into the ceiling and starts to walk out. The waitress says "Sir! You have to pay for your meal!" The panda hands her a badly punctuated guide and says "I'm a panda bear. Look it up." And the guide says "Panda bear: Eats, shoots, and leaves." -- In other words, he eats, then he fires a gun (shoots), and then he departs (leaves). It's a pun.