There are readily agreed upon sources and we have a relatively clear picture of the battle and how it progressed. One key source is Jean Le Fèvre de Saint-Remy's report on the battle, as he was a Burgundian noble who was actually present on the field. We also have an anonymous source, the Gesta Henrici Quinti, from the side of the English, believed to be a member of the royal household (maybe a chaplain or priest) who was present the day of, an anonymous source written by an unnamed French monk, and a relatively contemporary French history from Enguerrand de Monstrelet. The UK National Archives website actually has an entire section devoted to the Battle of Agincourt with records and accounts that you can read through.

In regards to the battle itself, there is some historical debate about the exact size of the forces involved, but general consensus as far as I am aware is that there were roughly 6k to 9k English troops present and probably about 4 times that number in terms of French troops. Historian Anne Curry has argued that the battle was likely 9k English vs 12k French, but given the lack of solid archival evidence one way or the other, most historians come down on the 4-1 ratio at minimum. Per the Burgundian accounts, Henry V did indeed deliver a speech to his men prior to the battle, though it was certainly not the "Band of Brothers" speech from Shakespeare, but rather focused on the validity of his cause under God. Multiple first and second hand sources from the English side confirm that Henry gave a speech. The Gesta actually quotes a line from the speech, a response to a knight asking about reinforcements, which goes, "the Almighty, with these his humble few, is able to overcome the opposing arrogance of the French." Not exactly Shakespeare, but full of confidence nonetheless.

As for the impact of the archers, it was undoubtedly significant, though dramatically increased thanks to Henry's decent battlefield selection, the relative overconfidence of French forces, poor leadership and organization on the side of the French, and the presence of terrain impediments for French forces attempting to attack the English formation. We know that the French cavalry, as they advanced to engage English forces, were funneled into narrow terrain over muddy fields while under a constant barrage of arrows that forced them to lower their visors and basically hunker down to avoid being wounded or worse. Many horses also died during this advance thanks to the arrows, impeding their riders and generally weakening the charge. As they slogged through the mud of the battlefield, their advance slowed, and when they eventually engaged with the center of the English line, the archers continued to pour arrow fire into the flanks of the attack at almost point-blank range, which was ideal for them to actually pierce armor and kill their targets. The archers then piled in from the flanks and pretty much overwhelmed the already strung out, disorganized, and tired French forces due to weight of numbers and sheer brutality in a nasty melee. The Gesta also reports that Henry himself engaged in hand-to-hand combat, protecting his brother Humphrey when he was wounded and narrowly avoiding death when an axe blow sheered off part of his helmet.

So, separating the myth from the battle. Well, surprisingly, the myth is relatively true to form based on everything we have. An outnumbered, tired English force under a dynamic military leader in the form of King Henry V met a larger, better provisioned, and better rested French force and beat the tar out of them thanks in no small part to the power and skill of their archers. Parts that I think usually get a bit left out of the myth is the sheer brutality of it all. Shakespeare touches on it, as does Kevin Branaugh's adaptation of Henry V, but the bloody melee were thousands of men died in agony, with some even suffocating to death in their armor due to the press of bodies, is not usually an overarching part of the legend. Keep in mind that, when the archers weighed in from the flanks, they did so not with bows or longswords or anything elegant, but rather with savage weapons made to kill me in armor: mallets, hatchets, falchions, cleavers, poleaxes (if they were lucky), daggers, and in some cases just clubs or other blunt instruments. Similarly, the state of Henry's forces and the desperation of the whole battle for the English (they really had no choice but to fight and certainly weren't in a good state to do so) is usually glossed over a bit as well. Of course, it's definitely Shakespeare play that pushes the Battle of Agincourt from a significant and laudable military victory into the realm of national myth. The adoption of Henry V as a patriotic piece to bolster English spirits during WWI and WWII essentially solidified the battle as part of the national mythos. Laurence Olivier's 1944 film version of the play in particular pretty much drove the nail home in that regard. It is also a good story, at least from the English prospective. People do, after all, love an underdog tale.

Sources
Agincourt: A New History - Anne Curry

1415: Henry V's Year of Glory - Ian Mortimer

Agincourt: Henry V and the Battle That Made England - Juliet Barker

The Little Grey Horse - Henry V's Speech at Agincourt and the Battle Exhortation in Ancient Historiography - Mogens Herman Hansen

National Archives: Agincourt - http://www.nationalarchives.gov.uk/agincourt/