It may sound too simple, but many of us forget how important it is to simply breathe. This is especially important for anyone with anxiety. Learning to breathe properly can help you relax as well as increase oxygen intake to all your vital organs and help eliminate toxins from your system.
Deep breathing is important in order to provide vitally needed oxygen to your brain and organs. Your body can live for weeks without food and days without water, but only minutes without oxygen. Your body also detoxifies itself by exhaling and deep breathing aids the body in clearing toxins more readily. Most people don't realize it but the lungs reach almost to the bottom of your rib cage and need to be fully inflated. All of the blood in the body exchanges oxygen for carbon dioxide in the lungs, so inhaling fully and deeply helps provide fresh oxygen to every organ in the body. It also forces the muscles of the chest and back to expand which helps you to relax. Muscles tend to contract when we are tense and this is the counter-movement to that tenseness.
It's important to note that deep breathing is not the same as rapid breathing. Deep breathing is slow and rhythmic. Rapid breathing is shallow and fast. Anxious breathing is also shallow and fast. Deep, slow, rhythmic breathing is the reason many people with anxiety disorders find yoga, meditation and even weight lifting to be helpful in managing their anxiety. All three, when done propertly, require the slow and thoughtful use of breath. If you are already hyperventilating due to anxiety, be especially aware of the emphasis on "slow" and "deep". You may even need to hold your breath for a few seconds on the inhale to get yourself slowed down.
Paul Ford has written an excellent article which describes the technique of "lung vacuuming" complete with diagrams.
If you prefer a video, on YouTube.com you can find Alexander Technique Instructor Eileen Troberman teaching how to inhale and exhale completely in a natural, flowing, relaxed manner. This technique is often used by singers.