The battle arena is now open! (And yes, you do have to participate in order to complete the bonus material.)
There are four people now outside the arena doors, but there are only two that you must talk to to finish the game - the man on the far left (Costume battle registration), and the woman on the right side (Bet on fights).
In order to complete an upcoming dungeon, you must first participate in the costume battle, and reach S-rank (which is the highest rank).
The game does not tell you that you need to meet this requirement - but if you reach the boss at the end of that dungeon and you haven't done the costume battle, he'll tell you he won't fight anyone who's not S-rank, and then you have to come back to Arena Town, do the battles, then go back to the end of the dungeon, which takes a really long time. (Don't ask me how I know this...)
Anyway, to do the costume battle, talk to the guy out front and choose "Challenge record." You need to challenge each rank in order, starting from Rank G and working your way up to Rank S.
Going through the Ranks
You need to win 2 battles at your current rank before you can challenge the next rank, and depending on your levels and your party setup, you may need to do some grinding or upgrade your gear before you can win the higher ranks.
When you challenge a rank, you'll have 3 choices: Challenge next opponent, Save current progress, or Abstain. You need to choose "Challenge next opponent" two times in succession, and beat both teams in order to move up to the next rank. Once you've got 2 wins, you can abstain, and challenge the next rank.
Note: If you defeat 1 team, you can save your progress if you need to, but when you reload you still have to challenge the next opponent. If you abstain after 1 win, you'll have to start over from zero.
Once you get 2 wins on Rank S, you can continue on with the game.
Bet on fights
The other thing that you must do here is to bet on fights.
This is quite a bit easier than the costume battles. Literally all you have to do is bet on which enemy team will defeat the other one. If your team wins the first match, you can rebet your winnings or you can cancel the match right there. Either way, you need to keep betting until you've received the Bronze, Silver, and Gold medals (these will be needed towards the end of the bonus material, but it just makes sense to get them now.)
Winning the Medals
To win the Bronze Medal, you need to bet on 10 fights and choose the winning team at least twice. You also need to win at least 1,000G off your bets.
To win the Silver Medal, you need to bet on 20 games and choose the winning team at least four times. You also need to win at least 3,000G.
To win the Gold Medal, you need to bet on 30 games and choose the winning team at least ten times. You also need to win at least 5,000G.
When you're done in Arena Town, warp to Forgotten Village and take the steps in the middle of town to your next destination - the Room of Flight.