The story is that the two-finger "V" comes from a French practice of chopping the index and middle fingers off the right hand of captured English archers. These two fingers were used to draw the bow and this mutilation would render them incapable of drawing a bow. English archers then responded by showing these two fingers to the French ahead of a battle in order to show that they were still capable of shooting.

The only credible (ish) source I've seen for this is Robert Hardy's book Longbow: a Social and Military History, but Hardy is an actor with a detailed enthusiasm for the longbow rather than a historian, the book in general offers a romanticised view of the longbow, and he cites no primary source for this claim.

As a Brit I like the story because it fits our national stereotype, but without any primary sources I am convinced that it is a myth.

The middle finger has been used as an obscene gesture since ancient times, due to its resembling a phallus with two testicles. Here's a passage about the philosopher Diogenes of Sinope from Diogenes Laertius' Lives and Opinions of Eminent Philosophers:

When some strangers expressed a wish to see Demosthenes, he stretched out his middle finger and said, "There goes the demagogue of Athens."

And a paragraph down from that:

Most people, he would say, are so nearly mad that a finger makes all the difference. For, if you go along with your middle finger stretched out, some one will think you mad, but, if it's the little finger, he will not think so.

He's writing in the 3rd century AD about a philosopher from the 4th century BC, so the gesture was understood in a similar way to today long before the Hundred Years' War.