According to some sources, by the time the French footmen reached English lines (the cavalry attack was of course a utter debacle) - they could barely swing their weapons from exhaustion. Perhaps the biggest effect of archery on armoured people was that they had to advance with visor down, trudging uphill through mud - visor down severely inhibits your breathing (which is why you generally put your visor up during melee). Of course, archers protected with wooden palings and firing from 3 sides prevented a direct cavarly assault.
Furthermore, they were too crowded - numbers don't help when all you get is people crowding you from behind (countless melee battles from ancient times to now, from Marathon, Cannae, etc have demonstrated this concept quite well) - the narrow field of battle prevented the French from using their numbers to outflank and overwhelm the enemy. Finally, the charge of the british archers vs their exposed flanks (they managed to push the english men at arms back somewhat) destroyed the french, which found themselves exhausted, huddled together and semi-encircled.
If you look how the battle unfolded, it was a classic move, the earliest recorded one about 1800 years old at the time of Agincourt: the enemy attacks the center, the forces holding the center fall back, then your flanks charge the now overextended and semi-encircled enemy. For bonus points deploy cavalry to cut off routing enemies, if it's possible / you have any available.
What really won the fight is English picking the ideal position and the French impetuously and foolishly attacking it, which is a recurring theme through that war. The English catastrophic defeats (like, eg. Patay) were also generally those where the opposite held true, longbowmen or not.
