You must kill High Chief Bristlelimb for the key that opens the cage.
Quest Text Edit
Help me, <name>! The Bristlelimb captured me and are planning to sacrifice me to their foul gods!
High Chief Bristlelimb holds the key that opens the cage. To force him to appear, you must slaughter the evil Bristlelimb furbolgs that inhabit the enclave, just two camps to the west of this camp. If you kill enough of them, he will surely come for vengeance! When he shows up, kill him and get the key!
My father will richly reward you if you can save me!
I don't have much longer, <Name>. They will kill me soon...
Once again you have proven yourself to be all that was prophesied. You have the eternal gratitude of the high chief <name> -- and these Stillpine relics to choose from...
The princess is at the most easterly of the 3 furbolg camps North East of Vorkhan the Elekk Herder. (Coords: 68,81)
You have to kill about a camp's worth of furbolgs at one of the two camps to the west of the princess. The high chief appears in the northwestern camp (coords: 64,77), so clearing that camp would seem optimal.
The high chief may already be around, as he appears after the furbolgs have been killed by anyone, not just people on this quest.