"Before going further though, you may have noticed that there are different types of them. Three types, I would say, but maybe I'm forgetting some.
The first type might sound irrelevant, but there's actually a significant number of arena quests that are accidental. You see, the activities performed in arenas are dangerous -- quite unsurprisingly -- which is why these areas are often built outside of town. But that makes them easier to invade by wild monsters… Which then become the targets of these quests.
The second type is for training and challenging Hunters. I have myself been to the arena many times as a young Hunter when I was learning how to use each weapon, and when I needed to sharpen my skills as I gained experience. After all, nothing makes you practice better than against a real Monster. Furthermore, the environment is safe enough for a newbie, and under enough control to allow comparisons between Hunters' performances.
I will admit I'm not sure how they get their hands on these Monsters however. I believe most of them are from hunter or poacher captures, usually the ones that are considered unable to be reintroduced in the wild (or to be used for research), so the Guild simply finds another use for them. I guess it can be debatable, though.
The last type I can think of is the most controversial of them all: arena quests for show; which aren't totally unlike the challenge quests from earlier. During hunting contests or simply for spectacle, arenas host an audience of people who want to see one or two experienced Hunters at work. The Monsters used for these quests are often more dangerous than average, so there's no way for them to be put back into the wild.
I... happen to know some Hunters who even specialize into such arena shows. One in particular, my cousin Magatsu, performs in Dundorma whenever he gets the opportunity. The few times I've seen him at work, the public seemed to enjoy his flashy moves, so I suppose it can be a popular 'sport'... To be enjoyed in moderation, if you want my opinion."