Jesus is introduced as "the son of David, the son of Abraham" (1:1). Remember them? Our good friends from the Hebrew Bible.
Matthew rattles off a long genealogy of Jesus with three sets of fourteen generations. Yeah, that's a lot of names.
After the genealogy breakdown, Matthew dives into the actual birth of Jesus. Don't worry, he leaves out the gory details.
Mary and Joseph are betrothed—that's just a fancy word for engaged. We happen to love that word.
But when Mary finds herself pregnant via the Holy Spirit (which would be understood at the time as God's spirit, not one-third of the Christian Godhead), Joseph decides he should quietly quit her before people find out.
An angel comes to Joseph in a dream and tells him to chill out. Why? Because Mary is pregnant with a son who will save people from their sins. Not only does this fulfill a prophecy (Matthew quotes Isaiah, for good measure) but it also sets the bar pretty high.
Joseph wakes from his dream a changed man. He takes Mary for his wife and doesn't have "marital relations" (adult speak for s-e-x) until after Jesus is born.