Instructions
Enter the Thunder Plains after reaching chapter five to start the optional mission "A Fallen Genius?" This requires you to defeat all of the enemies near each lightning tower.
Walk near each lightning tower as you travel north. This starts a fight with a Rhyos. You must fight nine Rhyos to make Humbaba appear.
Go to the spot the red arrow points at after you defeat all nine Rhyos. A prompt appears. Choose "Let's check it out" to start the battle with Humbaba.
Defeat Humbaba. Cast Protect on each of your party members to lower the amount of damage he can do. Lightning spells only do half the normal damage to him, but other elemental magic works normally. Humbaba casts Meteor right before he dies, so keep your party healed at all times to avoid being killed by this. Humbaba can be defeated instantly by using Eject or Bribe successfully. He does not cast Meteor if defeated with either of these abilities.
Walk back onto the spot you fought Humbaba to get the prompt again. It now contains a new option "Yup, in the hole." Use this option to enter the Thunder Plains New Cave. Cid and Machina Panzer are right inside the cave.
Defeat Machina Panzer. It has three weak Watchers helping it. Defeat all three Watchers before focusing on Machina Panzer. Cast Protect to decrease the damage done to your party by its attacks. Lightning spells heal it, but other elemental spells do normal damage. It is substantially weaker than Humbaba and defeating the Watchers significantly reduces its effectiveness. You are transported out of the dungeon after defeating Machina Panzer.
Return to the spot you fought Humbaba to re-enter the Thunder Plains New Cave. You can now explore it freely.