The Angel Wings are required to access the Castle in the Clouds, the only place in the game it is possible to obtain the Hero License (required for the Hero Job class). It can only be obtained by defeating Rico Jr., encountered as the boss at the end of chapter 4 and randomly after that. During the boss encounter, Rico Jr. has a 60% chance of dropping them. In random encounters afterwards, they have a 87.5% drop chance.
The Angel Wings are similar to the Passport. If the player lands on the entrance to the Castle in the Clouds without them, his or her turn will simply end. Also, unlike the Passport, the Angel Wings are a one-time use item.