To incarnate means to become flesh. The incarnation of Jesus is when the human nature (Jesus the man) was added to the nature of God the second person of the Trinity. It is where God became a man (John 1:1,14; Phil. 2:5-8). It was the voluntary act of Jesus to humble himself so that he might die for our sins (1 Pet. 3:18). Thus, Jesus has two natures: Divine and human. This is known as the Hypostatic Union.
The doctrine is of vital importance to the Christian. By it we understand the true nature of God, the atonement, forgiveness, grace, etc. It is only God who could pay for sins. Therefore, God became man (John 1:1,14) to die for our sins (1 Pet. 2:24) which is the atonement. Through Jesus we have forgiveness of sins. Since we are saved by grace through faith (Eph. 2:8-9), it is essential that our object of faith be accurate. The doctrine of the incarnation ensures accuracy--the knowledge that God died on the cross to atone for sin and that the God-man (Jesus) is now in heaven as a mediator (1 Tim. 2:5) between us and God.
Jesus came to reveal the Father (Matt. 11:27; Luke 10:22), to do His will (Heb. 10:5-9), to fulfill prophecy (Luke 4:17-21), to reconcile the world (2 Cor. 5:18-21), and to become our High Priest (Heb. 7:24-28). (Contrast with Kenosis)