Old English bræð "odor, scent, stink, exhalation, vapor" (Old English word for "air exhaled from the lungs" was æðm), from Proto-Germanic *bræthaz "smell, exhalation" (source also of Old High German bradam, German Brodem "breath, steam"), perhaps from a PIE root *gwhre- "to breathe; smell."
The original long vowel (preserved in breathe) has become short. Meaning "ability to breathe," hence "life" is from c. 1300. Meaning "a single act of breathing" is from late 15c.; sense of "the duration of a breath, a moment, a short time" is from early 13c. Meaning "a breeze, a movement of free air" is from late 14c.