Just as in XI, completing the final stage of the relic weapons will unlock the relic's weapon skill while that weapon is equipped on a character. Gate of Tartarus is uniquely treated as a spell though, using MP instead of SP (skill points).
Not all of the relic weapons are represented however. You can obtain the bow, sword, greatsword, shield, staff, dagger, and great katana. In order to help complete the relic weapons, you can trade currency, attestations, and fragments with other players in the event town's tavern. This is actually pretty unique for the game which is entirely single player and features very little interaction with other players beyond a friend list and messaging feature. Some currency, attestations and fragments can also be obtained with Gil (event currency) from the event exchange shop.

To fully complete the story you will need to play through the dungeon (at least) three times.