The reasons that nothing goes to waste in a forest are as follows-
Remnants of dead animals and plants get decomposed to form humus by the decomposers present in the soil. Humus acts as a reservoir of nutrients which increases the fertility of the soil and helps trees/saplings to grow.
Broken branches of the trees are used by hunters or people living there.
Thus, the waste generated in forest gets recycled and is used up again.
Dead animals are eaten by the scavengers living in the forest like vultures, jackals, etc.