In a book called Sefer Hasidim ("The Book of the Devout"), the 613 limbs and organs of the human body are associated with the 613 commandments of the Torah, each commandment corresponding to a specific part of the body.
Since each mitzvah gives off its own small light, man, then, is an array of lights.
Our lives consist of lighting one light here, one light there.
All these lights together make up the ideal image of the person, as though man were merely a brace, or a stand, for the 613 lights.
Man's duty is to make light.
Divine light illuminates the world through the flame of the mitzvot, while life on earth enables the soul to develop and go through a process of maturation.
We can now see why mitzvah and soul are equated: Just as the oil of the mitzvah is transformed into light, so man's soul becomes the light of God.
–Rabbi Adin Steinsaltz
From "The Soul of Man is the Lamp of God," an essay by Rabbi Adin Steinsaltz