When someone is angry, their nostrils flare out (think of the classic cartoon of the angry dragon breathing fire). That is the allusion of the phrase.
To expand the idea a little, the idea is that anger affects the rate of breathing. It's a well-known strategy that breathing slowly helps you stay calm. Breathing rate is perhaps a more integral part of anger than facial expressions, etc. since it's subconscious and not easily faked. Additionally, breathing rate is directly correlated to heart rate, but only the rate of breathing can be observed by an onlooker.
On a Kabalistic level, anger is said to affect the neshama, which is connected to the breath of the nose. (The books of Mussar bring down that when a person angers, his neshama is on some level replaced with an impure spirit). Hence, another preference for using this aspect of anger over others.