Why does the flow of energy through the biosphere
depends on the cycling of nutrients.

Since the transferrence is interrelated in a system called the food chain where the nutrients from plants is carried off by the first-consumers and second-consumers and so on untill the decomposers take the energy from the withered living organism and back into the system, it is mostly the transferrence of energy.
This flow of energy is present in its development and thus, the food composed by the photosynthetic autotrophs then is changed and took in by herbivores and then when omnivores or carnivores eat them it is translated into another form where it is stored in the cells used in many bodily activities.