On YouTube I can watch extremely detailed videos of ancient Roman battles that show what each section of the army did and at what time. How is this known, exactly? Was there just a scribe sitting on a hill recording the battle?

Well yes, there were always reporters for the Novum Eburicum Tempora.

But seriously, it is worth just looking at the source material for this. The reconstruction of Teutoberg is more complicated than most in part because batlefield archaeology has actually played a role in shaping our understanding of it, so instead I will use Cannae as one of the most famous battles of the ancient world and one that tends to feature in these sorts of battlefield reconstruction. From Polybius' Histories 3.113-116:

Next day it was Terentius' turn to take the command, and just after sunrise he began to move his forces out of both camps. Crossing the river with those from the larger camp he at once put them in order of battle, drawing up those from the other camp next to them in the same line, the whole army facing south. He stationed the Roman cavalry close to the river on the right wing and the foot next to them in the same line, placing the maniples closer together than was formerly the usage and making the depth of each many times exceed its front. The allied horse he drew up on his left wing, and in front of the whole force at some p281 distance he placed his light-armed troops. The whole army, including the allies, numbered about eighty thousand foot and rather more than six thousand horse. Hannibal at the same time sent his slingers and pikemen over the river and stationed them in front, and leading the rest of his forces out of camp he crossed the stream in two places and drew them up opposite the enemy. On his left close to the river he placed his Spanish and Celtic horse facing the Roman cavalry, next these half his heavy-armed Africans, then the Spanish and Celtic infantry, and after them the other half of the Africans, and finally, on his right wing, his Numidian horse. After thus drawing up his whole army in a straight line, he took the central companies of the Spaniards and Celts and advanced with them, keeping rest of them in contact with these companies, but gradually falling off, so as to produce a crescent-shaped formation, the line of the flanking companies growing thinner as it was prolonged, his object being to employ the Africans as a reserve force and to begin the action with the Spaniards and Celts.

The Africans were armed in the Roman fashion, Hannibal having equipped them with the choicest of the arms captured in the previous battles. The shields of the Spaniards and Celts were very similar, but they swords were entirely different, those of the Spaniards thrusting with as deadly effect as they cut, but the Gaulish sword being only able to slash and requiring a long sweep to do so. As they were drawn up in alternate companies, the Gauls naked and the Spaniards in short tunics bordered with purple, their national dress, they presented a strange and impressive appearance. The Carthaginian cavalry numbered about ten thousand, and their infantry, including the Celts, did not much exceed forty thousand. The Roman right wing was under the command of Aemilius, the left under that of Terentius, and the centre under the Consuls of the previous year, Marcus Atilius and Gnaeus Servilius. Hasdrubal commanded the Carthaginian left, Hanno the right, and Hannibal himself with his brother Mago the centre. Since the Roman army, as I said, faced south and the Carthaginians north, they were neither of them inconvenienced by the rising sun.

The advanced guards were the first to come into action, and at first when only the light infantry were engaged neither side had the advantage; but when the Spanish and Celtic horse on the left wing came into collision with the Roman cavalry, the struggle that ensued was truly barbaric; for there were none of the normal wheeling evolutions, but having once met they dismounted and fought man to man. The Carthaginians finally got the upper hand, killed most of the enemy in the mellay, all the Romans fighting with desperate bravery, and began to drive the rest along the river, cutting them down mercilessly, and it was now that the heavy infantry on each side took the place of the light-armed troops and met. For a time the Spaniards p285 and Celts kept their ranks and struggled bravely with the Romans, but soon, borne down by the weight of the legions, they gave way and fell back, breaking up the crescent. The Roman maniples, pursuing them furiously, easily penetrated the enemy's front, since the Celts were deployed in a thin line while they themselves had crowded up from the wings to the centre where the fighting was going on. For the centres and wings did not come into action simultaneously, but the centres first, as the Celts were drawn up in a crescent and a long way in advance of their wings, the convex face of the crescent being turned towards the enemy. The Romans, however, following up the Celts and pressing on to the centre and that part of the enemy's line which was giving way, progressed so far that they now had the heavy-armed Africans on both of their flanks. Hereupon the Africans on the right wing facing to the left and then beginning from the right charged upon the enemy's flank, while those on the left faced to the right and dressing by the left, did the same, the situation itself indicating to them how to act. The consequence was that, as Hannibal had designed, the Romans, straying too far in pursuit of the Celts, were caught between the two divisions of the enemy, and they now no longer kept their compact formation but turned singly or in companies to deal with the enemy who was falling on their flanks.

Aemilius, though he had been on the right wing from the outset and had taken part in the p287 cavalry action, was still safe and sound; but wishing to act up to what he had said in his address to the troops, and to be present himself at the fighting, and seeing that the decision of the battle lay mainly with the legions, he rode along to the centre of the whole line, where he not only threw himself personally into the combat and exchanged blows with the enemy but kept cheering on and exhorting his men. Hannibal, who had been in this part of the field since the commencement of the battle, did likewise. The Numidians meanwhile on the right wing, attacking the cavalry opposite them on the Roman left, neither gained any great advantage nor suffered any serious loss owing to their peculiar mode of fighting, but they kept the enemy's cavalry out of action by drawing them off and attacking them from all sides at once. Hasdrubal, having by this time cut up very nearly all the enemy's cavalry by the river, came up from the left to help the Numidians, and now the Roman allied horse, seeing that they were going to be charged by him, broke and fled. Hasdrubal at this juncture appears to have acted with great skill and prudence; for in view of the fact that the Numidians were very numerous and most efficient and formidable when in pursuit of a flying foe he left them to deal with the Roman cavalry and led his squadrons on to where the infantry were engaged with the object of supporting the Africans. Attacking the Roman legions in the rear and delivering repeated charges at various points all at once, he raised the spirits of the Africans and cowed and dismayed the Romans. It was here that Lucius Aemilius fell in the thick of the fight after receiving several dreadful wounds, and of him we may say that if there ever was a man who did his duty by his country both all through his life and in these last times, it was he. The Romans as long as they could turn and present a front on every side to the enemy, held out, but as the outer ranks continued to fall, and the rest were gradually huddled in and surrounded, they finally all were killed where they stood, among them Marcus and Gnaeus, the Consuls of the preceding year, who had borne themselves in the battle like brave men worthy of Rome. While this murderous combat was going on, the Numidians following up the flying cavalry killed most of them and unseated others. A few escaped to Venusia, among them being the Consul Gaius Terentius, who disgraced himself by his flight and in his tenure of office had been most unprofitable to his country.

From this account you can get most of the details you will see in videos like the one you posted: the weak Carthaginian center that fell back to allow the double envelopment of the Roman lines, the cavalry action on the sides, the screening by the skirmishers, etc. What you don't see is the conceptualization of the army into neat "blocks" that tend to dominate popular representations of battles. The Romans did organize their army in a way that could sort of allow that as the armies were dvided into named legions (~5000), cohorts (~600) centuries (~80) and conturbium (~10),^1 but it is very rare to see that reflected in battle literature unless a subdivision of the army did something remarkable (such as Cato's detachment during the Battle of Thermopylae). That is largely a modern convention, and someone more familiar with the development of modern military theory can probably comment on that better than I can.

^1 These numbers are very approximate and vary across time and space, and the conturbium may not have been thought of as a tactical unit.