The Sun is the center of our system, the planetary body around which all other bodies orbit. It is the life-giving force which, in astrology, represents our identity in most general terms. It is the Macro-manager of our lives; the part of our selves that makes the final decisions. It represents reason (not emotions, hopes, dreams or instincts).
As the Sun travels through the Zodiac (from our persepctive on Earth), the way it acts changes. Each sign gives it a different mask, a different filter through which it bestows its life-giving force on the individual. Ultimately, the influence of your sun sign on your identity is about the influence of the Sun's light filtered through the lens of the zodiac sign.
But what kind of light?
The daily light. The one that is everywhere, the one in which everything is soaked during the day. It is diffused, all-encompassing, it is the overall energetic field within which a baby is born. By comparison, the ascendent sign, which represents the sign that was rising on the Eastern horizon in the moment of birth, is like a thin ray of laser-focused energy on a newborn's forehead. It has a great impact because it is shining directly on th baby's forehead, but its impact is much more linear and therefore limited. The Sun provides the general energetic field of your birth.
You can find the influence of a person's sun sign in the overall structure of their life and legacy. You find the influence of their ascendent sign in the way they are in everyday life, who they are when they sit in a room and vibe. The ascendent is about expressing your self in the here and now. The Sun is about strategizing to build a life that reflects who you want to be.
This is why the Sun is associated with our creative powers - the power to create ourselves and our lives, the vision about who we are and who we want to be that we are here to manifest.
The house placement of the Sun gives us indication about how the individual will leave their mark on the world. Where they can find success in building the life they want. What area of life they should focus on in order to become themselves, in order to externalize their identity.
Ultimately, the Sun's role is to be the conductor of the orchestra. By virtue of its all-penetrating energetic influence, it harmonizes the influence of all other elements of the zodiac. It blends everything together. It serves as the gravity (scene, stage) of your identity that grounds the influence of all other planetary players over your life and identity. The Sun is the holding glue, the plasma that everything else swims in, and in order to swim in, it has to make itslef compatible.