We are programmed to be emotional creatures. Our reaction to our triggers is shaped by our personal history, experience, background, culture, and personality. Emotional intelligence is our ability to recognize these triggers and manage our emotions in a healthy way.
Self-awareness allows us to accurately perceive our own emotions and understand our typical reactions.
In his book, Emotional Intelligence 2.0, Travis Bradberry explains that the communication between your emotional and rational brains is the physical source of emotional intelligence. The rational area of the brain (prefrontal cortex) can't stop the emotion "felt" by the limbic system (the amygdala, regulates emotion, memory, and survival instincts) but the two areas influence each other and maintain constant communication. Because we were born emotional creatures, we tend to respond to events in an emotional way instead of a logical way. But we can control the thoughts that follow an emotion, therefore, the way we react.
Sources
Bradberry, T. Emotional Intelligence 2.0