Collin Hansen, Christianity Today:
Given the doctrine's [the Trinity] complexity, it's no surprise that we turn to analogies for help. But every analogy breaks down. "Most analogies drawn from the physical realm tend to be either tritheistic or modalistic in their implications," Millard Erickson writes in Christian Theology. Following Augustine's lead, Erickson therefore opts for analogies drawn from human relationships, though he admits that they, too, fail to convey the deep beauty of this central Christian confession.
"We do not hold the doctrine of the Trinity because it is self-evident or logically cogent," Erickson writes. "We hold it because God has revealed that this is what he is like."
This should be enough to answer our "So what?" question. We care about the Trinity because this is how God has shown himself to us in the Bible, even if we have to put the puzzle pieces together.
But we should also carefully study the Trinity in order to learn from this dynamic relationship. Modalism and Arianism misrepresent the wonderful fellowship within the Godhead from all eternity. The Trinity reminds us that God did not create us because he was lonely. God even draws believers into fellowship with himself through the work of Christ and the agency of the Holy Spirit.
The Trinity is no mere abstraction. It is God's plan of salvation in action. God the Father, desiring to restore fellowship between himself and his Creation, sent his incarnate Son, who willingly gave his life as a substitute for sins. After defeating evil by raising his Son from the dead, God sent his Holy Spirit as the seal of salvation for all who believe. So what? Nothing less than fellowship with God is at stake.