I've only played it on the Wii, but it does two things:

If you're collecting the pieces during the open-world between stages, you can save them up to buy new characters to use in open play.

If you're collecting them during the levels, if you collect it until the meter at the top of the screen is flashing, you'll be "True Adventurer". This gives you an extra mithril block at the end of the level to use in Bree for crafting items. Any coins you win during the level can also be applied to buying new characters.