That video is...not very good. The "units" and movements are pretty much totally made up--Caesar did not have two video game-style units of cavalry against four unspecified video game units of Gallic horse. Caesar gives no figured for either side of this particular engagement, although he says that Commius set out for Alesia with eight thousand cavalry and fifty thousand infantry--if we assume that the majority of Commius' cavalry were involved in this fight then we can make a conjecture as to the Gallic force. Have you read the de Bello Gallico? Caesar says right out what happened. The creator of that video has the wrong disposition of Gallic light troops. Commius did not station three neat little video game units of archers behind the cavalry force. Caesar describes the dispositions:

Galli inter equites raros sagittarios expeditosque levis armaturae interiecerant, qui suis cedentibus auxilio succurrerent et nostrorum equitum impetus sustinerent. Ab his complures de improviso vulnerati proelio excedebant.

Among their cavalry the Gauls had intermingled some archers and light-armed troops, who would provide support for their comrades if they fell back and take the charge of our troops. Some of our troops, wounded by these, fell back from this unexpected fight

There's no ambiguity in Caesar's language there. Not that it necessarily matters in the long run, as the result is the same, but it's just one case of the creator making questionable and wrong editorial choices. Another one is his bizarre decision that the Germans made up the entirety of Caesar's cavalry. That's simply untrue, and I don't see where he's getting this from. Caesar describes perfectly clearly what happened:

Cum a meridie prope ad solis occasum dubia victoria pugnaretur, Germani una in parte confertis turmis in hostes impetum fecerunt eosque propulerunt; quibus in fugam coniectis sagittarii circumventi interfectique sunt. Item ex reliquis partibus nostri cedentes usque ad castra insecuti sui colligendi facultatem non dederunt.

when they had been fighting from midday almost until sunset, with victory in doubt, the Germans in one part ["of the line" should probably be understood here, or perhaps one part of the plain or the hill--either way Caesar means that the Germans only made up a part of the force] bringing together their squadrons charged the enemy and drove them in a rout. And when those had been sent into flight the archers were surrounded and killed. Likewise from the other parts [of the field, or the line, or whatever] our troops [he does not say if this includes the infantry as well, nostri doesn't tell us much] pursued them all the way to their camp, and did not give them the change to rally themselves

In any case it's pretty clear what happened. In a closely-contested, uphill battle, the numbers of which are not known (Caesar may give a figure for his cavalry somewhere in the previous book, but I don't remember him doing so--in any case he gives no figures for this fight at all), the Gauls, tired out by hours of fighting, were set upon by a massed column of German horse, which managed to punch through the line and panic the force, at which point the rest of the Roman line fell on the enemy.

There's a pretty good museum near what is generally agreed to have been Alesia, near Alise-Sainte-Reine. For a long time the location of this battle was uncertain, until Napoléon III funded an archaeological dig that suggests we have the right location. They have a lot of information about the lives of Roman soldiers, what they could hope to earn to take back home, how big the units were and so on. I'm afraid my memory fails much of that detailed information on unit size. Indeed, the numbers involved are difficult to know.

Our primary source for describing the battle is Caesar's own Commentarii de bello gallico ('Commentary on the Gallic Wars'). His numbers might be inflated for the purpose of self-aggrandisement - though the victory is impressive by the most conservative estimates. Caesar states the 80,000 men were besieged at Alesia, a town that was well fortified atop a hill, and his own force was plainly unsuited to taking it. The goal was to defeat Vercingetorix and thus take the figurehead of Gallic opposition to Rome. Caesar's response is singularly genius and mindbogglingly ambitious - to construct a perimeter wall around the entire area, and starve out the Gauls.

What about Caesar's army made this feat possible, and what about the Gauls made it not? The walls' construction having begun, Vercingetorix and his cavalry seek to disrupt the Romans, perhaps hoping to prevent them gaining a strong foothold. The speed and force of cavalry is excellent for sorties from a defensive position - a force can slip out quickly, engage, and retreat to relative safety. The Romans engage the Gallic cavalry, and the outcome seems to have been claimed by Roman discipline. Infantry stood fast at the Roman camp, whilst the cavalry engaged, secure in the knowledge that they could fall back to this line. Caesar claims the enemy's number worked to their disadvantage, limiting their mobility as they retreated, funneled into the city gates. Caesar praises his German contingent for fearlessly pursuing the enemy even to this bottleneck, diminishing their number greatly and causing a panic in which many soldiers left their horses to flee. Caesar claims that Vercingetorix sent the last of his cavalry out to find a relief force, escaping between the gaps in the incomplete siege wall. He quotes Vercingetorix as laying the lives of 80,000 at stake for their mission, referring to the besieged force.

A theme that Caesar returns to again and again is the idea of his forces' discipline. Roman infantry were skilled craftsmen, and felling a woods to construct palisades, walls, guardtowers, moats, and traps was something he could expect them to do quickly and efficiently. The panicked Gauls could do nothing but watch as the walls went up. The walls faced both inwards towards Alesia, and outwards towards any reinforcements. They were also compartmentalized, with internal walls between sections, such that if any section were to fail, it would be contained. Romans would have time to gather in adjacent sections and retake it - like a bulkhead system on a modern ocean ship.

The men of the town sent out their women and children, also the sick and the infirm of the town to Caesar, in order to stretch food rations, expecting Caesar to let them leave - instead, Caesar allowed them to starve, unless Vercingetorix surrendered. By this time, a relief force was within sight.

The genius of the defensive system was put to the test. A frontal attack from without was rebutted. The Gauls attempted again to attack a section by night. The Gauls had to focus effort on one section for their numbers, reported by Caesar to be 250,000 (certainly less, but still greater than the Roman number), to be useful. The circular nature of the Roman fort meant that cavalry could leave the unassailed sections of the wall and come at the force from behind, attack, and then return into the wall again. By daylight, the Gauls found themselves having made little progress and surrounded by the threat of Roman sorties, and retreated. Attack from within and without was difficult to coordinate, and by the time Vercingetorix and his men were in a position to fight, impeded by the complex web of traps, they were too late to divide Roman attention. Gallic attempts to defeat the Roman position were systematically controlled, to the point that Vercingetorix finally offered his surrender.

In sum, Caesar had disciplined soldiers, competent subordinates who understood his ideas to lead units (Brutus, for example), he chose an ideal form of engagement for his unit size and the enemy position, and he had time on his side with Vercingetorix and his men set to starve.

Primary source: Julius Caesar, Commentarii de bello gallico