The Totomostro is about one thing, and one thing only: bloodsport! Yep, cruel though it may be, you can gamble on the prowess of a team of monsters, which will then fight to death for you amusement. Gamble correctly and you'll win medals, your winnings will vary depending upon the odds assigned to the monster team you bet on (which varies by the type of monster(s) that comprise that team). You can buy medals at the rate of 10 gil per medal, and you can bet up to 9,999 medals in a single match. Considering that some items you can redeem your medals for cost up to 3,600,000 medals... that can take an insane amount of gil, so you'll have to put in the time at the arena is you want the best prizes.
So, buy a starter fund of medals and gamble yourself up until you can start dropping the max medals down on each fight. You can't save/load in the arena itself, but there's no shame in securing your initial investment by save/loading between fights to ensure you're not flushing gil down the drain. After one or two successful fights at 9,999 medals, you should have enough medals to cover any infrequent losses.
Of course, the only way to keep your losses infrequent is to know what monsters to bet on, and this is part common sense, part learning. Some monsters are just far stronger than others - you can usually take a Coeurl to the bank. In general, you should disregard the level of a monster, as it rarely means much in terms of what will win the fight. Something that does matter, however, which varies battle by battle, is each monster team's condition. This is denoted by a series of blue arrows near the monster team's odds in the "Wager" menu. A low condition monster isn't as strong as a high condition equivalent, and this can sway thing just enough to cause some upsets.
What monsters you should pick largely depends on the type of monster (Coeurls and Iron Giants are usually pretty safe bets), their condition, and what and how many monsters they're fighting against, with the first two factors being the most determination of success. Unfortunately, luck also plays a role in the outcome of battles - if your high-condition Gigantoad decides to hop into the middle of two groups of fast, little critters that focus their attention in it... well, you might not be happy with the outcome. Generally, all condition values being equal, in a one-on-one fight you should pick multi-animal teams (especially Sabertusk variants), as they're likely to gang up on the lone opposition and defeat them. In multi-team battles, endurance becomes more important (especially when you bolster your monster with the Horn of Fortitude).
Aside from learning which matchups work well for your medal collection, the only way you'll participate actively in the arena is via horns. Randomly during bouts you'll get a prompt to blow your horn - mash the Circle/B button to fill up the meter and grant your horn's effect (the Horn of Fortitude will see you through the arena just fine). If no monster team wins in the time limit, the match will end in a draw and you'll get back whatever you bet... basically just a waste of time.
Below is a list of the items you can win at the Totomostro:
|Mythic Color Samples||360,000|
|Big Master Typhon||45,000|
|Great Garula Tusk||2,500|
There are a few unique items you can redeem here, and when you get them, they'll be replaced by other items that you can buy multiple times. For example, once you buy the Llymlaen reel it'll be replaced by a Platinum Bangle. As you can probably guess, some of the top prizes in the arena will take an absurd amount of time to earn, and it's worth pondering... is it worth it?
If you do obtain the Magitek Generator (which removes the need for fuel for the Regalia) it'll be replaced by an Onion Bangle, a very nice item that gives the massive boost of +2,500 HP. Of course, it's going to cost 3,600,000 medals for the Magitek Generator, then 2,400,000 medals for each Onion Bangle. The latter might be worth your time, but the former? You'll be saving at best hundreds of gil with the Magitek Generator, and once you upgrade the Regalia to the Regalia Type-F, you'll have unlimited fuel anyways. Plus, 6,000,000 medals for an Onion Bangle is far more time-consuming that just playing 10,000 gil Justice Monsters V for the same prize.
For the sake of completionism, it might be worth grabbing an Enforcer and a Precision Lance, but otherwise, unless you have to grab every decal and accessory for the Regalia, just do enough to get the Llymlaen reel, which will make finishing Navyth's fishing questline a bit easier.