Respiration is the process which leads to breakdown of food and release of energy (and not breathing, although breathing is a component of it). During the aerobic respiration (process involving oxygen, as happens in human beings and many other life form), food is first converted to glucose and then this glucose reacts with oxygen. This reaction results in release of carbon dioxide and (most importantly) energy in the form of ATP (adenosine triphosphate) molecules.
All the life processes, whatever we do, require energy. We use energy for walking, eating, talking, running, sleeping, thinking, writing, reading or any process that we do or is required for our survival. Same is true for other life forms. And that is why respiration is important, because it provides the much-needed energy for our survival and growth.
Hope this helps.