Why you should raise monsters in your mog garden.

If you raise a monster in your mog garden, you can unlock cheers of that monster. Cheers are passive and constantly active bonuses to you and your trusts.

Every cheer has a different bonus, one of the best cheers being from the Jumbotender monster (evolved from Pequentender - rank 7). This cheer, when unlocked gives:

Trusts: constant Refresh +3 to all trusts, regen +5, Capacity bonus +7%, Move speed +3%,

https://www.bg-wiki.com/ffxi/Monster_rearing