Your metabolism is tied inextricably to your respiratory system, because the respiratory system is responsible for bringing in the oxygen you need to burn nutrients for energy, and for clearing out certain metabolic waste products. You can't affect your metabolic rate through your respiratory system, however; rather, the former affects the latter.
Colloquially, metabolism refers to the production of energy by burning nutrient molecules -- and the expenditure of that energy through a variety of processes, including cell maintenance. However, from a biochemical perspective, your metabolism is the sum of all chemical reactions taking place in your body, not all of which relate to energy production and expenditure. Still, it's the energy production portion of metabolism that's most intricately tied to the respiratory system.
You need oxygen to burn most nutrient molecules, explain Drs. Reginald Garrett and Charles Grisham in their book "Biochemistry." Technically, it's possible to burn sugar to produce very small amounts of energy without using oxygen -- this is called anaerobic metabolism, and it's what your cells do when you engage in a very hard effort -- but for the most part, you rely upon aerobic metabolism. This process requires oxygen, which the lungs provide.
When you inhale, you bring in air, which contains oxygen. The oxygen diffuses across the thin cell membranes of the lung tissue into the bloodstream, where proteins called hemoglobin in red blood cells pick it up and deliver it to the tissues. Respiration serves a second purpose, however, explains Dr. Lauralee Sherwood in her book "Human Physiology"; it helps you rid your blood of carbon dioxide, which is what you produce when you burn nutrients in oxygen.
When your cells burn more nutrients for energy -- as they do when you're working out, for instance -- you require more oxygen, and produce more carbon dioxide, which you must get rid of. This causes you to breathe faster. Conversely, when you're burning fewer nutrients for energy -- as when you're sleeping, for instance -- you don't need as much oxygen, and are producing less carbon dioxide. The lungs respond by working more slowly.