When you Center yourself, you are aligning yourself to Source and the Earth. The idea is to focus your energy, to pull that energy into alignment down through the center of your body, and typically to focus a gathering of energy in one area of the body with intention.
In moving meditation, such as Tai Chi, you pull your energy down into the don tian, the Sacral Chakra, located below your navel. This is the lowest of the energy centers (for this practice) and is meant as a temporary storage space for your energy. During practice, you move from this center. You keep your balance low.