To perform this quest, you must first save the Frogsmith. He's located south of the Village of Outcasts. The Titan's Mitt is required to release him from his prison. If you have the Magic Mirror, you can walk back to the location of their house, Mirror back, and enter their home.
Once he rejoins his partner, you need to leave the house, re-enter, and pay 10 Rupees to receive their item.
Upon completing that quest, the Purple Chest becomes available.