Bear in mind these are dependent beings....not unlike a child or elderly person. They both have rights as well, but we don't 'let them go in the wild'. (Queue jokes.. ;) We care for them, as they're unable to care for themselves.
The point of the lawsuit is not as focused on current animals (all of whom are dependants), but more-so for future generations (free-living/wild chimpanzees). If granted personhood, they wouldn't be used in the first place, and animal sanctuaries would be needed less, and demand for them (poaching, breeding) would fall. This way, they wouldn't be in zoos, in labs or other exploitative environments, and instead would be where they ought to be - in the wild.