The underbelly of something is its secret weakness, something about it that's not protected from harm. Once a computer hacker discovers a network's underbelly, they're easily able to hack into it.
Literally, an underbelly is just that: it's the abdomen of an animal, the soft lower belly. Often, that's also the most vulnerable part of an animal's body, since it contains major organs, and it's where the figurative meaning comes from. An alternate informal definition is "a hidden, criminal part of society," like the underbelly of a city where criminals trade stolen goods.