This usage often relates to natural or energy cycles, stressing interdependence in ecosystems. The forest ecosystem feeds off a balance between predators and prey.
Pro Tip 2/3
Metaphor Alert
'Feed off' in this sense can be metaphorical, describing a process rather than physical eating. He feeds off positive feedback to boost his performance.
Pro Tip 3/3
Emotional Fuel
It's often used to discuss things that provide emotional or spiritual energy. Musicians sometimes feed off the crowd's energy during a concert.