The Shroom is the mutated form of Palet after he consumed sap taken from Yggdrasil after being caught by Ryu and the others. Palet then takes the form of a huge mushroom with feet and a large red and blue mushroom cap that looks similar to a brain.
Shroom is a very well-rounded boss and by far the toughest at this point, so exploiting weaknesses is difficult. Fortunately, Shroom has very little to no counter rate, so that makes the battle easier. It's recommended that Rei or another character with Steal + Charm is in the battle in order to steal the Wisdom Fruit early. After that, the best tactic is to have Ryu use ample amounts of his Shield spell, followed by Momo using Might several times. If Nina is in the party, she should focus on using her best spells (if she has them, Magic Ball and Mind Sword can be very helpful here, especially in conjunction with Meditate). If she obtained Shield from Yggdrasil, she should spend a few turns using that. Her naturally high Agility can also allow her to use healing items before Shroom can attack if needed. If Rei is in the battle, avoid using Weretiger until closer to the end, as he goes berserk a bit more quickly in this battle for some reason (though if someone has the Influence skill, then that problem will be removed, allowing him to run rampant). Garr should stick to physical strikes, particularly after using Focus, where one blow can deal 350 or more damage without Might, depending on his level. Ryu should not transform, instead he should focus his AP to healing and buffing, as it takes priority, especially if Shroom uses Blitz or Ragnarok, which can deal impressive damage. Simply put, physical strikes boosted by Momo's Might can deal around 200-300 points of damage a shot, depending on the character, while group-based healing techniques are necessary to survive the battle. He will use Restore when he starts wearing down, but it only restores about 200 HP, which is probably the equivalent of Ryu or Momo attacking once, so it only stalls the battle slightly. Just keep attacking, healing when needed, and Shroom will go down.