Matthew's primary audience were the Jews. He starts this passage with Jesus' genealogy to get their attention and help them relate to Him. For the Jews it was important to prove the lineage of the royal line from David, it was a qualification for the Messiah (true king of Israel).
Jesus fulfills the prophecies of the Old Testament, the Hebrew Bible. Christmas is about what God has done, it proves that God is a promise keeper. He kept his promise to Abraham that his descendant would bless the nations.
Four women (plus Mary) were mentioned in Jesus' lineage. These women were associated with some messy issues, which also highlights the need for a Messiah to atone and redeem us from our sins.