They cornered them using a combination of mounted chasers, dogs, and pre-placed cages/nets manned by brave (or foolish) attendants. There were professional hunting agencies (venatores) who accepted and executed contracts, and also purchased pre-captured beasts from enterprising groups on the fringes of the Roman world (like nomadic groups passing between Roman North Africa and the deep interior). During the Republic, at least, up-and-coming senators depended on their contacts in the provinces to secure beasts for their games. Marcus Caelius Rufus, for instance, repeatedly asks Cicero to help him acquire leopards while the latter was on official business in Cilicia.

There are several very famous ancient depictions of hunts (the gathering of the beasts, and not to be confused with the "Hunt" itself, the venatio, in the arena on the day of the games).

Getting a rhino, North Africa

An animal cage, from the famous series of Late Roman mosaics from the Villa del Casale

Showing mounted chasers, a wheeled cage, and soldiers on foot

Things didn't always go according to plan.

Chasers, herding, and the nets, from North Africa

More from North Africa.