Breathing deeply is a refreshing way to calm yourself down, relax, and bring yourself into the present moment in competition. The present moment is like a portal to being "in the zone". Moreover, taking some calm deep breaths decreases tension muscles, reinvigorates your cardio, and actually is a bit of a boost in your confidence levels. The idea behind the rise in confidence is that the deep breath calms you and when you're calm you're in a more confident state. You don't need to be tense when you're walking in the park because it's easy as pie. It's not something that you are fearful or anxious about engaging in. There is complete confidence that you can walk through a park to the point that you do it without thinking. Breathing deeply can take you on subconscious level to a state such as that.
Another good blurb about breathing deeply comes from Experience Life Magazine:
Developing mental toughness begins with shutting out negative thoughts. Most sports psychologists agree that the best way to block unfavorable thoughts is to focus on breathing deeply.The brain can think of little else when focused on that one act. “You no longer focus on the distractions,” says Kenneth Baum, author of The Mental Edge (Perigree, 1999).