Three major elements.

First, they survived the political crisis brought on by the defeat. They replaced the dead senators, elected new consuls, raised new armies and kept most of their allies and subjects from defecting. So while the defeat was costly it did not succeed in breaking Rome's internal order or its diplomatic position.

Two, they learned to deny Hannibal the decisive battles that he was so skilled at, and instead to go on the offensive against less-able commanders in Spain (and Macedonia, when they briefly entered the war on Carthage's side). The Fabian strategy frustrated Hannibal and turned the war in Italy into a long, slow-burning conflict that nullified Hannibal's advantages.

Three, the Carthaginians made critical mistakes that gave Rome breathing room after Cannae (Hasdrubal lingered in Spain for unknown reasons instead of reinforcing Hannibal as requested) and created opportunities for them to counterattack (the Carthaginians later left Carthago Nova almost completely undefended in the face of Scipio Africanus' army in Spain). Without these missteps Carthage would have had a very good chance of victory in the long run.

Reference: Mastering the West: Rome and Carthage At War by Dexter Hoyos