What does Matthew 1:3 mean?

Matthew is establishing the line of Jesus from Abraham. This establishes that Jesus is the fulfillment of God's covenant with Israel to bless all the nations. It also proves He is the fulfillment of God's promise to David that his descendants would always sit on the throne of Israel. Matthew began in familiar territory for all Jewish people. Abraham was the father of Isaac, who was the father of Jacob, who was the father of twelve sons, the fourth-born of which was Judah.

Now Matthew follows the line through Judah, who was the father of twin boys named Perez and Zerah (Genesis 38:27–30). These boys were born to him as the result of a sordid sexual encounter with his daughter-in-law Tamar, described in Genesis 38. Judah failed to follow through on providing the widowed Tamar with a husband, according to the levirate marriage custom. So she disguised herself as a prostitute and became pregnant with Judah's child.

Tamar is the first of five women mentioned in Matthew's genealogy of Jesus and not nearly the only person connected to questionable moral choices. This is significant. First, genealogies did not require the mention of any women, at all, since the line was passed through male descendants. Matthew goes out of his way to mention these women, including a conception under questionable circumstances

However, this fits into the focus of Matthew's book: Jesus, who came to offer grace and forgiveness from God to humanity through His own life and death and resurrection. It also fits with the counter-cultural way in which Jesus honored and esteemed women during His ministry on earth. As someone once involved in a profession considered despicable—a tax collector for the Romans—Matthew would have appreciated God's ability to use sinners to accomplish His will.

Matthew continues following the line from Judah's son Perez to his son Hezron (Genesis 46:12) and to his son Ram (1 Chronicles 2:9).