There are several reasons why the incarnation of Christ is vital to the whole of Christian belief, but there are two major reasons.
Firstly, Christ becoming flesh gives a cogency to God's redemptive plan, which is first alluded to after the fall in the third chapter of Genesis. Christian doctrine is built upon the idea that humanity has a sinful nature that has no desire or ability to have community with God on its own. If that premise of fallenness is true, then God's plan must include his own work to bring his people back to himself. After all, a holy God cannot ignore the justice that humanity's sin deserves, nor will a loving God abandon his work in the hearts of his people. As the book of Hebrews says: "Indeed, under the law almost everything is purified with blood, and without the shedding of blood there is no forgiveness of sins" (Hebrews 9:22, ESV). According to Christian doctrine, the incarnation of Christ is both part of God's plan to fulfill his righteous wrath (by pouring it on his son) and a demonstration of his love for humanity and his dedication to them for his own ultimate glory. Peter says in his first sermon in Acts that "this Jesus, delivered up according to the definite plan and foreknowledge of God" was crucified for sin, thus validating the idea that God's ultimate plan was always to sacrifice his own son as propitiation for sin (Acts 2:23a, ESV).
Secondly, the humanity of Christ exemplifies the depths of love in the Christian God for his people in a personal way that goes beyond salvation. One of the most difficult issues to wrestle with in life is the problem of evil. Theists as well as non-theists struggle with this issue, but Christian doctrine states that Christ suffered evils, including temptations, deaths of family and friends, and his own torture and murder. Therefore, the Christian faith's belief in a Savior who suffered for his people gives a personal testimony to the fact that, although suffering occurs in the world, it cannot be that God does not care for or about that suffering. He even subjected his own son to that suffering for a period of time.