An expression made famous (and controversial) by Heineken and their openly gay spokesperson, Neil Patrick Harris, which pokes fun at the sanctity of a man's barbecue and (presumably) his heterosexuality.
According to Neil Patrick Harris, Heineken Light "makes it OK to flip another man's meat" to which the griller replies that no man can do that. (Should include syntactical variations of 'Flipping Another Man's Meat'.)
It means you want to engage in sexual acts with the person. Instead of only using your middle finger to give someone the bird, adding your thumb let’s said person know you want to get it on with them.
Kyle: *Flips girl off with thumb*
Girl: “He flipped me off with his thumb. He wants to fuck! Flipping someone off with your thumb means they want to do it!”