The answer to why and how we are able to feel more energized after being active is dependent on your anatomy as well as physiology. To be more precise, the organelles (e.g., mitochondria, nucleus, etc.) within your cells adapt to the training stimulus. In this case with regard to energy, it is the mitochondria (known as the "energy storehouses") of your cells which increase in number when your body becomes more active. Take, for example, weight training. As you perform more and more weight training exercises, the mitochondria within your muscle cells begin to multiply. This causes an increase in mitochondria density which creates more potential for energy to exist. It is the mitochondria that is responsible for up to 95% of your body's energy level.
So, to summarize, as you exercise regularly, your body reacts by increasing its storage of cellular mitochondria which creates more energy. In effect, you need to exercise to feel more energy. If you don't exercise, there's no need for you body to increase its mitochondria supply and as a result, you'll fell lethargic and weak. It's a classic case of "if you don't use it, you'll lose it". If you don't exercise, you'll lose your energy capacity (less mitochondria density). What irony there is in that to feel more energized, you need to begin exercising rather than relaxing. The more active you are, the more energized you'll feel. It's that simple. Of course your energy levels also hinder on your nutrition so be sure to eat healthy to complement your training. Nutrient deficiencies can disrupt cellular function and impact your energy as well as performance.