A pouch is a small container or bag, like the pouch full of dog treats you carry when you take your new puppy to obedience school. Sit! Stay! Good boy.
It used to be much more common for people to carry things around in pouches for safety and convenience — money pouches, mail pouches, tobacco pouches, and so on. Often these were drawstring bags that cinched securely at the top. The word came to be used for other things with a similar shape, including animal cavities meant for carrying babies (think a kangaroo's pouch) and various pocket-shaped receptacles.