Losing a fight or going 60 days without fighting in the arena will lose you ranks.
See the Arena Rank page for details.
Winning in the arena obviously depends upon getting exceptional pets and giving them good equipment. A silver eyed witch will do well for her level, especially when equipped with some artifact gear. Bells, meanwhile, are spectacular gladiators when properly equipped. Use Gene Engineering to give Marksman and Eye of Mind to a silver bell, and they will train it rather rapidly in the arena.
The pet arena can also be a good place for the pets themselves to be trained in their martial skills, provided the arena doesn't produce monsters too tough for them to handle, and their potentials aren't terrible - a ranged attacker whittling away at a mithril golem while staying out of melee range, can get off hundreds of attacks before that golem goes down.
Level of opponents scales with the rank, when rank is maxed out, and with current win streak.
You'll receive 100 Yen coin every 5th win and 500 Yen coin every 20th win. In addition, for every 100th win in a row, you'll receive 20000 fame and every pet will be rewarded with 2000*win_streak gold, capped at 600000 gold pieces.