Outer breathing refers to the breath that keeps the body alive. Inner breathing refers to that which gives life to the lower self or the Higher Self (the original spirit). What we experience as 'I', as our identity, is the result of the thoughts and emotions we have. Even our memories are thoughts and emotions that occur in the present moment.

All that we are is the result of our thoughts. -- Buddha

When an impulse in the heart called intent (see chapter 2 #7), creates a movement of energy resulting in an emotion (e-motion or energy motion, called qi (气) in Chinese) and a corresponding thought, this causes a feeling of 'I'; this feeling of 'I' is the lower self. When there is a true intent in the heart to turn around the light and as a result of this true intent, the Higher Self awakens, the inner breath now gives life to the Higher Self. Esoteric traditions call this breath the Breath of Life. The breath of life is not a physical breath, but an energy motion that gives life to the Higher Self or Original Spirit.

Ra is the divine god, who created himself, who made the heavens and the earth, and the breath of life.
-- Egyptian Legends of Ra and Isis
The breath of life is the consciousness of life.
-- Upanishads (Hindu text)
There are six cosmic breaths; they are matched by the six breaths in man. -- Book of Salvation of the World by the Manifestation of Dao from the Supreme Concourse (Taishang huadao dushi xianjing) The World Upside Down by Esabelle Robinet

Hathor, giving the breath of life (symbolized by the Ankh) to the Pharaoh (Egyptian Museum, Cairo)