A combination of factors conspired to prevent the French from being able to fully leverage their numerical advantage against the English.

One of the first things to examine is the battlefield itself. You can see from this diagram the general shape of the battlefield, although this specific arrangement of forces is controversial. Note how the terrain effectively creates a choke point that 1) prevents the French from enveloping the English forces and 2) creates a narrower front for the English to focus their archery on. Henry V was an experienced commander well-acquainted with the strengths and weaknesses of his army, and he had picked his ground accordingly. Before the battle began, he had his archers prepare pointed stakes to be hammered into the mud in order to ward off French cavalry. Battlefield fortifications of this type were an extremely common tactic of the period. When these preparations were combined with carefully selected terrain like at Agincourt, a clever commander could give his side an enormous advantage.

This brings us to the second point: longbows. The extremely large numbers of archers the English brought to Agincourt is probably the most famously explanation for the English victory. However, this was not a unique factor. English armies had maintained disproportionate ratios of archers to men-at-arms for the past century or so. While a longbow arrow shot at maximum range will generally not typical plate armor of the period, they were very capable of causing a wound at the weaker joints of the armor. Many French knights were killed by arrows striking them in the face when they lifted their armored visors for the sake of vision. But that's somewhat besides the point. The primary purpose of massed archery volleys was not necessarily to cause large casualties, but to disrupt enemy formations and harry their advances or charges. Instead of attacking the English in large waves, the French footmen and knights who had slogged uphill through the mud and hails of arrow fire were attacking in smaller knots of men.

Because of the above factors, while on paper, the French had a numerical advantage, that was not necessarily the case locally. Outnumbered French knights or men-at-arms might find themselves being swarmed by a group of English troops, clubbed over the head, disarmed, and dragged back behind the English lines to be ransomed later. This was another advantage of the English deploying so many archers: their light equipment allowed them to move more easily through the muddy field, and come to the assistance of the heavily armored English men-at-arms in the melee.

That's all tactical discussion, though. Strategically speaking, the French also faced difficulties. They had initially begun gathering troops in order to end the siege of Harfleur, but the town fell before they could rally enough men. They would continue to shadow the English on their long march from Harfleur to Calais, constantly increasing in numbers and strength. As I discussed previously in this comment, the initial French plan of attack was far more cautious, calling for the deployment of skirmishers to soften up the static English formations and a hand-picked task force of knights to clear the English archers from the flanks. But as more nobles and soldiers arrived in the French camp, command and control broke down. Arguments over rank and experience, as well as simple confusion as to how to deploy so many additional men, caused the abandonment of the initial battle-plan. The skirmishers were shuffled towards the back of the ranks, as men-at-arms crowded forward to join the vanguard. By contrast, the English were a much smaller ( and therefore much easier to control) force led by experienced veterans and captains, with an energetic and popular king in command. They experienced none of the command and control issues that plagued the French army, and were relatively well-disciplined by comparison.

With all of the above considerations in mind, it is perhaps not surprising that such a catastrophic French defeat occurred at Agincourt. After all, the English had delivered other such victories on previous occasions (Crecy, Poitiers, Najera) and both sides were well aware of their respective strengths and weaknesses. Contrary to depictions in Shakespeare's Henry V, the English army was not really a starving band of peasants outgunned by an overwhelming opponent. It is true that dysentery was taking a toll on Henry's army, but the fact remained that that army was an extremely capable, battle-hardened force led by competent and experienced officers. As I noted in this comment, the longbow archers of Henry's army were in large part representative of local economic elites from England's towns and villages, not drafted peasants, and many were professional soldiers in their own right. It could be said that "sheer numbers" seems to have been the only advantage the French possessed at Agincourt.