The screen shows 3 different bars of life for 3 different targets in this battle - the left lung (red), right lung (grey blue) and core (in the center, protected by a heavy shield).

The left lung absorbs red and green orbs left by enemies, and the right lung absorbs white orbs and your Devil Trigger.

The point is to take out the life of the core, which will show itself for a bit only after you take out either of the two lungs. Then the core will close and the lung will be regenerated. So, this is the plan (Reading may be confusing):
 

1. Switch to Swordmaster and equip Rebellion and Agni & Rudra (a.k.a. A&R). Air Hike for both or either of them is optional but it'll be a lot easier if you have it.
    
2. At the start of battle, run to the right lung, lock on to it, (double) jump and press Circle. Regardless it being Sky Dance (A&R) or Aerial Rave (Rebellion), once done with the move (make sure you're still in mid-air), switch weapons quickly using R2 and press Circle again. Keep switching weapons and press Circle as you stay in mid-air attacking the right lung.
   
   Note: Ignore the left lung. You will never need to touch it.
    
3. When you feel you're about to fall on ground, jump off the heads of the Hell Envies below you to keep yourself in mid-air and keep slashing the lung. If you fall to the ground, jump/roll away ASAP 'cause those enemies are SO ready to attack you. Then repeat steps 2 and 3.
   
   Note: There is also another way to do this, but it may be a little harder to grasp. If you're interested, check out the mission guide for mission 8.
    
4. The core, however, won't make your life that easy and will release orbs at you. They're slow but they hurt. Stop the slashing cycle and wait for the light to come close, then run away and around, then back to the right lung.
    
5. Try not to stand where the orb will hit the enemies 'cause they die instantly when hit and they may drop green orbs, red ones as well, that will be sucked away by the left lung, which will heal the core. (This occurs mostly in the harder modes)
    
6. Once the right lung is done for, the core will open up, finally. Run to it and start attacking with either weapon. Close your distance quickly using Stinger, but be careful not to press Triangle when close to the poles (when trying to perform Stinger) or you'll end up wasting your time doing a pole swing instead.
    
7. The Hell Envies will follow you to the core to attack. If you're patient enough, forget about the core and get away from them cause' trying to attack them can be rather risky – all it takes is for one of them to hit you and the rest will follow up. If you're impatient, you can try attacking and forcing them away with A&R as you continue to hit the core – not recommended for harder modes.
    
   Devil Trigger Explosion (DTE) tutorial:
   If you have 6 or more DT slots (make sure it's full), you can make use of DTE to get rid of all those Envies very easily. Press and hold L1, and you'll notice that the DT slots starting from the 4th will blink (in red) one by one. The more blinking red slots, the stronger your DTE. You can do this anytime, even as you're attacking, evading, running, etc.
    
   So, begin charging up DTE as you're dealing with the right lung. By the time the core opens, you should have long charged your DTE to the max. Keep L1 held down and attack the core until the group of enemies close in on you. When all of them are close enough, release L1. What happens is that you will activate Devil Trigger, but you will lose all but the first 3 DT slots. In return, the impact of your release will massively damage all who're close to you. Assuming you're right by the core as you do this, your DTE will kill all the Hell Envies and damage the core, giving you all the time to attack the core undisturbed.
    
8. Once the core closes, its next attack is likely to be the lasers. Notice how how the shield that usually covers the core shifts upwards so that the core can reveal itself to shoot the lasers? Strangely enough, in this time it IS possible to inflict damage upon the shield, which shares the same health as the core. So, target the shield, Air Hike and slash away at the shield the same way you do for the right lung. The lasers won't get you so long you keep yourself in the air. It's only a matter of time before the battle is over.