Impassibility is the doctrine which states — not that God has no affections, no “emotions” — but rather that, “God is not affected or changed by anything outside of himself. The main point is that God is not affected rather than lacking affection!”
Michael Bird then goes to explain how impassibility makes the gospel good news:
Baddeley goes on to argue that impassibility is what makes the gospel good news. First, impassibility proves that God acts out of love and in grace because it is only his own being that motivates him to be so. God is not persuaded to be loving and gracious because of his experience of suffering. Rather, God acts lovingly and graciously toward us because he eternally and unchangeably is so.
Second, the incarnation and cross were not simply a manifestation of what was always true, namely, that God suffers and empathizes with human tragedy. No, the incarnation was a fundamentally new event whereby the Son became incarnate and suffered for us. Therefore, things previously not possible for God’s Son, such as the experience of suffering and death, became possible when the Son took on flesh and blood. Impassibility establishes that God relates to us in a fully emotional way, grounded in his own nature; accordingly, he is able to act in love and grace toward us. Furthermore, the man Jesus Christ can authentically sympathize with our weaknesses because only God cannot be emotionally pressed to acting in compassion toward us.
So, then, does God suffer? Indeed he does. But his suffering is not surprisingly imposed; it does not move him to be something other than he is or to do something other than he intended to. God chooses to be the God who suffers with and for human creatures. Indeed, according to Packer, “a totally impassive God would be a horror, and not the God of Calvary at all. He might belong in Islam; he has no place in Christianity. If, therefore, we can learn to think of the chosenness of God’s grief and pain as the essence of his impassibility, so-called, we will do well.”
Bird, Michael F. (2013-10-29). Evangelical Theology: A Biblical and Systematic Introduction (Kindle Locations 2754-2765). Zondervan. Kindle Edition.