The story
From the advertising slogan It’s finger-lickin’ good adopted by the U.S. fast-food restaurant chain Kentucky Fried Chicken (now KFC) in 1956, referring to the fact that something eaten with the hands is so delicious that one cannot help but lick one’s fingers to savor the crumbs of food or juices left behind.
Etymology from Wiktionary, CC BY-SA 4.0.