What is Christmas all about?

It's about the incarnation, the Word becoming flesh, God becoming man.

But why do that?

There is only one thing God couldn't do without becoming incarnate in a body. Die.
Athanasius said:

The Word perceived that corruption could not be got rid of otherwise than through death; yet He Himself, as the Word, being immortal and the Father's Son, was such as could not die. For this reason, therefore, He assumed a body capable of death.
On the Incarnation chapter 2 paragraph 9.

Why die?

This saying is trustworthy and deserving of full acceptance: "Christ Jesus came into the world to save sinners" — and I am the worst of them.
(1 Timothy 1:15)

To save sinner, Jesus had to die to stop death winning, fulfilling the law and carrying it's curse in our place, offering himself as a sin offering to propitiate (turn aside God's anger) for our sins. It had to be a human body, as it was human flesh that needed saving:

Thus, taking a body like our own, because all our bodies were liable to the corruption of death, He surrendered His body to death instead of all, and offered it to the Father.
On the Incarnation chapter 2 paragraph 8.

[God] condemned sin in the flesh by sending His own Son in flesh like ours under sin's domain, and as a sin offering, in order that the law's requirement would be accomplished in us who do not walk according to the flesh but according to the Spirit. (Romans 8:3-4)

There are other reasons why the incarnation is great:

1. God is able to sympathise with our weakness as he's been tempted (see Hebrews 4:15)
2. We can know what love is:

   God's love was revealed among us in this way: God sent His One and Only Son into the world so that we might live through Him. Love consists in this: not that we loved God, but that He loved us and sent His Son to be the propitiation for our sins. (1 John 4:9+10)

3. That the Devil might be destroyed.

   Now since the children have flesh and blood in common, He also shared in these, so that through His death He might destroy the one holding the power of death—that is, the Devil
   (Hebrews 2:14)

4. We need not fear death
   

   Now since the children have flesh and blood in common, He also shared in these, so that through His death He might destroy the one holding the power of death—that is, the Devil — and free those who were held in slavery all their lives by the fear of death.
   (Hebrews 2:14+15)

5. To give us an example of true humility
   

   Make your own attitude that of Christ Jesus: who, existing in the form of God, did not consider equality with God as something to be used for His own advantage. Instead He emptied Himself by assuming the form of a slave, taking on the likeness of men. And when He had come as a man in His external form, He humbled Himself by becoming obedient to the point of death—even to death on a cross.
   (Philippians 2:5-8)

6. God is fully made known through the person of Jesus:
• We have an image of the invisible Father, that we can see, in the Son (see Colossians 1:15)
  

  No one has ever seen God. The One and Only Son — the One who is at the Father's side — He has revealed Him.
  (John 1:18)
  if we know the Son, we know the Father
  (John 14:7)

• We see God's Glory

  we have seen his glory, glory as of the only Son from the Father, full of grace and truth.
  (John 1:14)
  For God, who said, "Light shall shine out of darkness" — He has shone in our hearts to give the light of the knowledge of God's glory in the face of Jesus Christ.
  (2 Corinthians 4:6)