A chinese girl

A sigh is an audible exhalation of one's breath, and is often a means of expressing an emotion. "To heave a sigh of relief" means to exhale audibly, for the purpose of expressing one's sense that something has ended, or one has escaped something, which might have been bad, or at least worse than it was in fact.

Before you do something difficult or energetic, you often take a deep breath.

When you heave a sigh of relief, you inhale and exhale, "Whee! That's over with."