Hail the heaven-born Prince of Peace, Hail the Son of Righteousness
Piggybacking on the previous stanza, this stanza again emphasizes that Jesus is from heaven. He comes to earth from heaven because he is God. He is the Prince of Peace, an allusion to another fulfilled prophecy found in Isaiah 9:6. The Prince of Peace in Isaiah is also called "Mighty God." No doubt about it, the Prince of Peace is God. Thus, he can truly be deemed righteous. He is the son of righteousness, that is, all that is righteous proceeds from him and is in him. He is truth and morality and holiness.
Light and life to all he brings, Risen with healing in his wings
Now we get to the clincher--the why this king, Messiah, Christ Jesus has come. The first stanza hinted that he would reconcile man and God, but how? This stanza says that he will bring light and life. That he is risen is a hint to his death. This Messiah will die and yet rise again and come with healing for man in his wings. As for the wings, I can't help but think of Jesus crying out as he came into Jerusalem, "O Jerusalem, Jerusalem, you who kill the prophets and stone those sent to you, how often I have longed to gather your children together, as a hen gathers her chicks under her wings." Indeed, he would rise showing that healing could only come to those who gather themselves under his wings.
Mild he lays his glory by, born that man no more may die
This healing that will come is not for temporary sickness, but for the pervasive sickness that leads all men to death: sin. In humility, the Messiah left heaven for earth (already described in the second stanza) so that man would no longer die. If he will no longer die, then this must mean the Messiah will solve man's sin problem.
Born to raise the sons of earth, born to give them second birth
Here then is the crux: Through Christ's own death and resurrection, man can himself die and be raised to life. Man can receive a second birth into life devoid of sin and live eternally. I am reminded here of the conversation between Jesus and Nicodemus: "In reply Jesus declared, 'I tell you the truth, no one can see the kingdom of God unless he is born again.' 'How can a man be born when he is old?' Nicodemus asked. 'Surely he cannot enter a second time into his mother's womb to be born!' Jesus answered, 'I tell you the truth, no one can enter the kingdom of God unless he is born of water and the Spirit. Flesh gives birth to flesh, but the Spirit gives birth to spirit.'" Through rebirth through Christ, man lives again one with the Spirit. His sin problem and thus death is conquered. This, then, is the truth of Christmas. We celebrate the incarnation because it leads to our freedom from sin and death. The true joy of Christmas rests in our own salvation through the sacrifice of the incarnate God.