They commissioned teams of professional hunters. Sometimes, if especially large games were in the offing, Roman troops would help.

The methods varied with the animal and region. Pliny the Elder, for example, tell us that pitfalls were used to catch individual elephants, and that teams of mounted hunters sometimes drove whole herds into narrow canyons, trapping them there until they were weakened by hunger and could be led away (HN 8.8).

As for lions, Pliny tells us the following (HN 8.21):

"It was formerly a very difficult matter to catch the lion, and it was mostly done by means of pit-falls. In the reign, however, of the Emperor Claudius, accident disclosed a method which appears almost disgraceful to the name of such an animal; a Gætulian shepherd stopped a lion, that was rushing furiously upon him, by merely throwing his cloak over the animal; a circumstance which afterwards afforded an exhibition in the arena of the Circus, when the frantic fury of the animal was paralyzed in a manner almost incredible by a light covering being thrown over its head, so much so, that it was put into chains without the least resistance..."

Tigers were (supposedly) captured by a very different method. A hunter would steal a tigress' cubs and ride off with them on a fast horse. The tigress would pursue; but whenever she got close to the hunter, he would drop a cub, forcing her to stop and return the cub to her lair. She would then resume her pursuit of the hunter - but by then he would have already regained the safety of his ship, with at least one of her cubs (Pliny, HN 8.25). In his Hexameron (6.4.21), St. Ambrose describes the same method, but says that the hunter drops a mirror instead of a cub: the tigress, seeing her reflection in the mirror, thinks she has found her cub, and paws at the mirror while the hunter escapes.

The famous Mosaic of the Great Hunt in the Villa Romana del Casale near Piazza Armerina in Sicily (see pp. 32-3 of this PDF for an overview) shows various African animals being transported to Rome for the games. Both hunters and armed Roman soldiers take part. A leopard is lured into a trap baited with a live goat; a tigress is baited with a glass ball (a version of the mirror trick described by Ambrose). Animals are loaded into carts, and finally led into ships with large crates on their decks for the journey to Rome.

Those intrigued by the beast hunts that took place in Rome might be interested in my page on the Colosseum.