The psychic influences of the moon on our lives are numerous and often subtle. This is quite a wide subject, so let's focus on some general motives that can give a starting point to understand the spiritual side of our moon.
We are living on one celestial body (Earth), we get light and energy from another (Sun), and there's another one orbiting above our heads (the Moon). Compared to these masses our human bodies are puny - basically, these celestial masses present the context of our lives which we take for granted.
They define a kind of rhythm which we live by: one year is the time the Earth goes around the Sun, one day is the time the Earth turns once around its axis, and a lunar month is the time the Moon goes around the Earth. This kind of rhythm is deeply buried into our bodies and the way we perceive reality. A normal person doesn't consciously pay attention to this rhythm, but a psychic does (or at least should).
As far as psychic concepts go, we usually see Earth as a nurturer and the Sun as a giver of life. The moon's role is not so clearly defined. One one hand it can block the light of the Sun during an eclipse and can represent darkness. On the other hand, during nights when there is no direct sunlight, it act complementary to the Sun and reflects its light to us indirectly.
In general, the psychic moon can represent rhythm, darkness in light, light in darkness, a balance between life and death, and balance between sure knowledge and the unknown.