Meownster Hunters is a way for your cats to go out and gather materials on your behalf. You won't unlock it at first so do village quests until d'Artanyan lets you go. It's a really fun and (somewhat) easy to play minigame that can land you gatherable materials, monster materials (sometimes), and the all-useful scraps.
You unlock more areas as you unlock more areas to hunt in. More stars = better chance for more loot. Warning indicates that lots of big monsters are running around and Rare indicates a high chance of getting rarer, more valuable rewards. Once you select an area, you can select up to four Palicos to go. In order to launch them off, you use a cannon. Totally not animal cruelty!
The circles the cats zoom through as you launch them are circles the cat will be able to gather from. But ultimately, the circle the cats land in will most greatly affect what you get as rewards. If a rare area shows up as you're using the cannon (it fades in when you're above it but not any other time) try your best to aim for that area to land your cats there! Note that your cats can rebound off the edges of the screen, so if an area seems hard to get to, try higher power and a different angle to bounce your cat there. Then do a quest or a few quests and the cats will return.
Note that cats with very low Enthusiasm cannot embark on the Meownster Hunters.