Costumes are a feature in the game that allow you to wear other pieces of equipment (e.g. equipment that is mostly for looks but have poor stats) over your existing equipment (e.g. equipment that may not look great but have good stats). Like many other role playing games, these things exist but for a price (fortunately, unlike most role-playing games you do not need to pay real currency, but rather gold that you can obtain through many different ways in-game). Remember that you cannot obtain stats or effects (including set effects or enchants!) from a costume.
The more equipment you have, the less bag space there is for you to collect other important materials or quest items. It is important to balance out the equipment you want to dedicate to making a costume set for yourself, and equipment which is used to strengthen you for later battles.
They are unlocked by purchasing a Prismatic Cloth from Andy's shop in Ashaya Village for 100 gold (90 with Revered status). Using the piece of cloth in the inventory will use it up, permanently enabling the option to use costumes in place for all of your equipment.