Collectible Synthesis unlocks via a short "talk to me" style quest in Mor Dhona. At that point, you can start crafting and handing in Collectible versions of other items that you would normally craft, similar to Leves, except you can hand in an unlimited number (depending upon time and inventory space), and you cannot HQ a Collectible.

From 56 through 59, Collectibles that you hand in not only give you Gil and Exp, but also Blue Crafting Scrip. At 60, you can also start earning Red Crafting Scrip. Scrip can be used to buy various things, or exchanged for Tokens, which can be used in other purchases.