It certainly wasn't easy.

The Romans relied on a vast network of professional hunters, trained soldiers, and hired help to track, net, and transport the thousands of animals that appeared in the Empire's arenas every year.

I talk more about the hunters in this answer.

The default hunting method was to use teams of beaters (usually men with torches) and dogs to chase animals into netted enclosures. To capture big cats, a deep hole was dug, and some small tender thing (usually a kid goat or puppy) tethered to a post or pillar of earth in the center. When the tiger or leopard attempted to pounce on this morsel, it fell into the pit. Elephants were usually chased into blind valleys, and penned there until hunger made them docile. Ostriches - so we are told - were lassoed by men on fast horses.

For more detail (I apologize for the short answer, but I'm a little pressed for time), I refer you to my video on how animals were captured for the Colosseum. Feel free, of course, to ask about any particular of the process you want to know more about.