Instructions
Proceed through the game until you've read all of Chico's briefings. The last of his briefings entails you giving him a photograph of a UFO. Once you've done this, you'll gain access to the "Monster Hunter" levels.
Open the Mission Selection Menu and select Extra Ops 29. This lands you on a remote beach. Nearby, you'll spot a cat. Approach it and it will talk to you. After the conversation, you'll fight your first "Monster Hunter" beast.
Kill the monster. If you're lucky, it will make a drop. If not, you'll have to try again. You can speak to the cat to fight this monster as many times as you want. Each time you complete a "Monster Hunter" mission, you also unlock additional missions and more monsters to harvest.