All Mickey has to do is destroy/befriend all the enemies in the area. There are two Hoppers in the area, as well as a Basher and seven Spladooshes. The Beetleworx have to be eliminated, but the Blotlings can either explode or be painted.
If Mickey deals with the enemies, he will just have to talk to Starkey again to complete the Quest. Starkey will exit with him. When the mouse goes back to Tortooga, the area will be enemy-free. Mickey can talk to Starkey to obtain the Pirate Hero Pin.
If Mickey doesn't complete this Quest, Starkey will stay trapped in the Jungle and Tortooga will be overrun by a Basher and some Spatters.