Grasping soft, irregular material is challenging both for animals and robots. The feeding systems of many animals have adapted to this challenge. In particular, the feeding system of the marine mollusk, Aplysia californica, a generalist herbivore, allows it to grasp and ingest seaweeds of varying shapes, textures and toughness. On the surface of Aplysia's grasper is a structure known as the radula, a thin flexible cartilaginous sheet with fine teeth. Previous in vitro studies suggested that an intrinsic muscle, I7, is responsible for opening the radula. Lesioning I7 in vivo does reduce opening width, but does not prevent animals from grasping and ingesting food. New in vitro studies demonstrate that a set of fine muscle fibers on the ventral surface of the radula, the subradular fibers (SRFs), mediate opening movements even if the I7 muscles are absent. Both in vitro and in vivo lesions demonstrate that removing the subradular fibers leads to profound deficits in radular opening, and significantly reduces feeding efficiency. A theoretical biomechanical analysis of the actions of the subradular fibers suggests that they induce the radular surface to open around a central crease in the radular surface and to arch the radular surface, allowing it to softly conform to irregular material. A three-dimensional model of the radular surface, based on in vivo observations and magnetic resonance imaging of intact animals, provides support for the biomechanical analysis. These results suggest how a soft grasper can work during feeding, and suggest novel designs for artificial soft graspers.