This is the standard simplified narrative of what happened at Agincourt - English archers wiping out French noble heavy cavalry - although it also seems like that was an exception.
The battlefield killing was done by light infantry wading in with daggers and hatchets apparently during a foot slog up the muddy hill which left the french heavy infantry exhausted. Another wave of killing afaik was when the captured prisoners were all executed since the english position seemed like it might be overrun by some follow up fighting.
(keep in mind the battle took hours and there was a lot more going on then just heavy horse riding up against arrows once or twice)
Wiki page is worth a read actually
First, that archers can actually be more effective against mounted troops than foot: the mounted troops ride close together, horses are hard to fully armor, and one horse getting hit in the leg can cause a lot of chaos.
Second, at Agincourt, the French knights _walked_ through the arrow-fire quite successfully, but the effort (physical, mental, cumulative effect of small wounds) tired them enough that the English soldiers could beat them hand-to-hand. And that this ability to inflict small damage before the main fighting is why archers were valuable.