In the first verse of the “Shema,” the Bible clearly states that there is but one God, who is in himself one. The eternally self-existent "I AM" (YHWH), the Creator of heaven and earth is one, but he has further revealed himself as embodying within himself the principles of relationship and association entailed in being Father, Son and Holy Spirit. God is a trinity, a unity of three persons. It is his very nature.
In some respect, we can “reverse engineer” the nature of humankind to understand the perplexing, trinitarian nature of God. We were made in the image of God, and so we are reflective of his nature, at least in some key ways. For instance, I am a father, but I am also a son, and there is a sense of me, a “vibe” let's call it, that is accessible to those around me. I am a finite and flawed reflection of the infinite God--Father, Son and Holy Spirit--in this respect.
Furthermore, when God looked at Adam, alone in the Garden, his response was that it was not good for man to be alone. Why? Could it be that since God in himself is not alone, and we are made to be like him, it was not good (i.e reflective of God) for man to be alone either? No man is an island, he was never intended to be. Humankind is made in the image of a triune, relational Godhead, and are not what they should be out of fellowship with others of their kind.
God's desire is that we would reflect him, not only in ourselves, but also with each other as Jesus stated in such soaring language in his “High Priestly Prayer.” The Trinity is not just an arcane church doctrine hammered out so long ago: it’s a practical understanding of God, and by extension, us. The relational Godhead has called us into relationship with him, like him. Father, Son, and Holy Spirit is more than a formula for baptism, it’s the fundamental nature of God, and it is the relational fabric we’re being woven into as the children of God.