It completes somewhat of a circle in God, the smallest there is in a "group." Another way of describing it is that Love manifests Himself as a person. So the Love of God the Father to the Son is by the Holy Spirit, and the Love of God the Son to the Holy Spirit is by the Father, and the Love of God the Holy Spirit to the Father is by the Son. Three is actually the "simplest" one. It is by which Love is known. Between two people, love is a mere energy, and yet this energy is not even known, and just trusted. With the Trinity, Love is Manifest and perfected in what we weakly call "Persons."
Any other attribute of God is made up of threes as well. Lover, Beloved, and the Life of Love. Enlightener, Enlightened, the Life of Enlightenment. And each take these roles together. With Humanity, it is necessary that we take the role of the Son, being enlivened by the Spirit, that through the Spirit, we become sons of the Father directly. And so we are part of the Trinitarian relationship in a mysterious way be being the "beloved" and the "enlightened" by grace.