The purpose was mainly to use them for the entertainment of citizens at spectacles, not only at Rome but throughout cities in the Mediterranean World. They generally would have been killed in animal hunt games. We have a series of letters between the famous orator Cicero and an ally of his, Marcus Caelius Rufus, from when Cicero was governor in Cilicia in c. 51 B.C. In these (for example, Cic. Fam. 8.2) Caelius Rufus consistently asks that Cicero get panthers for him for the games he was putting on as aedile; it was particularly important to do so because Rufus had competitors with other wild animals, and the crowds loved the exotic. Cicero was not pleased and continually told Rufus that there were no panthers in his province.

Over four hundred years later, in 401 A.D., the Roman aristocrat Symacchus was engaged in the same practic when he put on games in honour of his son's praetorship. Now, the stakes had changed by this point: Someone like Caelius Rufus was hoping to please the people so they would elect him to further offices. Symacchus, on the other hand, was engaging in the practice of euergetism: That is, using his wealth to show that he was a lover of his city and to receive honours and acclaim from the people in exchange and in the process, introduce his son in style. I'll quote the great historian Peter Brown's description of the lengths Symacchus went to to secure animals for his son's games, since in his characteristically delightful prose he brings together so many sources and explains what was at stake:

"Above all, the animal world poured into the midst of the city. And what an animal world! Symmachus went out of his way to bring them all to Rome. Crocodiles from the Nile and bears from the Balkans; great Irish wolfhounds brought all the way from Britain in iron cages; lions from the southern mountains of north Africa; antelopes and gazelles trapped along the edges of the Sahara. There was no frontier of the Roman empire not represented by this parade of exotic beasts. Their sheer diversity amounted to a celebration of the immensity of the Roman world. Like the empire itself, their capture and eventual slaughter represented a triumph of human order over a savage world." -Peter Brown, Through the Eye of a Needle, p. 116

(Incidentally, Symmachus shows how hard this was to pull off, even 450 years after Cicero's time: Some of his crocodiles starved on the way to Rome, only a few bear cubs showed up, and Symmachus had to petition the emperor to send him some lions).

And again, these games happened all over the empire, though the most lavish ones would have happened at Rome and wherever the emperor(s) was. But it's true that not all wild animals were imported for the games. They could be imported for other events as well, such as triumphs: Plutarch (Pomp. 14.4) tells us that Pompey brought elephants back to Rome for his triumph in 63 B.C. and that he wanted his triumphal chariot to be drawn by them. Sadly for Pompey and happily for the elephants, they would not fit through the gates and he was forced to use horses instead. But the same mentality was at play here: Exotic animals represented the exotic places that Rome had conquered. And if you could bring an animal that had never been seen before, that was even better, as Caesar showed when he brought a giraffe (or as the Romans called it, a cameleopard) for the first time to Europe for his triumph in 46 (Plin. NH 8.69). Finally, wild animals could be put on long-term display, as Augustus did (Suet. Aug. 43.4). So, not all wild animals were killed for the games. But the vast majority would have been.