So Hannibal's famous crescent formation did have the effect of drawing the Roman infantry in. But something I generally notice left out of the popular narrative is that the Roman infantry actually nearly broke completely through the crescent. In fact the Roman infantry drove so deep into the crescent that the Carthaginian line broke in some places. However, neither of our main sources for the battle, Polybius or Livy mention a complete break, so its probable that the retreats, if they did happen, were controlled and probably directed by the Carthaginians to further draw in the Romans. Its worth noting Hannibal had positioned himself in the centre, and thus would be there to direct affairs/motivate the men personally. I feel its important to note this, because its important to realize the Romans were assured of their victory basically until the Libyans hit their flanks. The Carthaginian infantry in the centre always seemed on the verge of breaking, and may explain why the Romans were caught so of guard.

Anyways the Romans drew so far into the crescent that it started to actually level out, that's when they discovered the Libyan spear men on their flanks. The Romans had been unaware that the Libyans had been outflanking them, most likely due to the presence of a Carthaginian skirmish line.

What the Roman infantry did next is best described by Polybius:

no longer kept their compact formation but turned singly or in companies to deal with the enemy who was falling on their flanks

So basically the Roman infantry panicked and abandoned their trademark discipline and formation to fight the approach Libyans. Just at this moment Hasdrubal led his cavalry into the Roman rear, greatly increasing Carthaginian morale. Again Polybius described it:

delivering repeated charges at various points all at once, he raised the spirits of the Africans and cowed and dismayed the Romans

After that it was only a matter of time, as disorganized Roman units were picked off one by one. I'll again cite Polybius (last time I swear):

The Romans, as long as they could turn and present a face 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 were all killed where they stood.

Now what's interesting about Cannae is that it seemed to fly in the face of what we know about Roman infantry. Namely Rome's main tactical advantage had always been that there units were flexible and individual leaders like Centurions, etc. Could take charge of the situation and lead. Think the battle of Cynoscephalae where individual Roman units waited for gaps to open in the Macedonian phalanx, before rushing through to strike at the Phalanx's exposed rear.

So the question is why didn't that happen here? Why didn't the Romans regain their individual composure and cut their way out? Well its never really explained in any ancient source, so we are left to speculate.

Basically it seems as if it was a perfect storm, the mixture of relatively fresh infantry in the form of the Libyans attacking the Roman flank, repeated cavalry charges preventing any sort of retreat back out of the killing zone, and finally Hannibal's skirmishers were no doubt having a field day with the packed in Romans; when you are trying to regroup, the last thing you want is a constant barrage of stones and javelins. Add in that the experienced Roman commander, Lucius Aemilius Paullus was killed in the fight, leaving command solely to Varro, who was nowhere near as experienced. So it seems as if the Romans were just never given the chance they needed to effectively regroup.

Now it should be noted some Romans did escape, namely the commander Varro. Some via cutting their way out, others probably just managed to escape in the confusion of battle. Another notable survivor was a young Scipio Africanus, who would later defeat Hannibal at Zama.

If you are interested in sources, I would start off by recommending the Ancient sources. Livy's The War with Hannibal and Polybius' work on the battle, which appears in "Book III" of his histories.

In terms of modern sources I recommend Cannae: The Experience of battle by George Daly and Ghosts of Cannae by Robert O. Connell.

A significant number did in fact punch through the centre of the Carthaginian line and escape. Livy gives a total of 19,000 who escaped in total.

As for why the Romans just didn't punch through everywhere - ancient combat wasn't like a RTS game where a general could just order his units to charge into battle and fight mindlessly. Morale, fear, tiredness all contribute a huge factor. Those soldiers on the flanks must have been surprised when they went from victory in pushing back a (seemingly) defeated enemies to being attacked in the flank. Turning to face them would have disrupted the formation (since your best fighters were in the front) and pinned that unit in place. Confusion would be rampant as men deeper in the formation wondered why they weren't advancing and officers would be trying to sort out their formations, try to get the men organized instead of milling about and panicking. Livy says the men got themselves packed too densely to properly fight - totally believable if the flanks recoiled from the enveloping attack - which would compress the Roman formation. This would hinder attempts to fight, maneuver units, and sow panic.

Looking at a diagram of the battle, we get a perfect birds eye view of what was happening. But as a soldier on the ground, you'd just see what was in front of you in an environment filled with the sounds of war, dust raised by movement, a wall of the enemy, while hiding behind your shield.

I'd like to ask a followup, if I may. /u/Sid_Burn talked about Hannibal being in the center, so that he could direct the affairs and /u/Agrippa911 spoke about denseness. What is the scale of the battle? I've seen lots of battle diagrams before, but I've never gotten a sense of sizes. About how big of an area would 85,000 Romans get packed into? About how long would Hannibal's lines extend in order to complete the maneuver?