Can you explain how matter and energy are continuously transferred between organisms and their physical environment?

All processes that exist obey the Law of Conservation of Mass and Energy. These laws state that mass and energy can neither be made nor destroyed. Thus, they are only transferred from one system to another. The more technical term is that, mass and energy can be transformed. Energy can be in different forms: kinetic and potential energy. 

Mass transfer happens when, for example, a deer eats grass. The grass contains mass and energy from the soil that gives it nutrients. When the deer gets eaten by a lion, the mass and energy transfers to them, as well. The cycle continues in the ecosystem.