What is the significance of Matthew starting with the names of the ancestors of Jesus?

The Gospel of Matthew begins, "The book of the generation of Jesus Christ, the son of David, the son of Abraham."

Thus, this is what is intended to immediately capture the reader's attention. And, why is this?

To Abraham, God said (Gen. 22:18),

And all the nations of the earth shall be blessed in your seed because you obeyed My voice.

The seed of Avraham in which all the nations of the earth would be blessed was Jesus Christ (Acts 3:25-26; Gal. 3:16).

To David, God said (Ps. 132:11),

...Of the fruit of your body will I set upon your throne.

The fruit of David's loins was Jesus Christ (Acts 2:30).

Therefore, Matthew intends to establish the fulfillment of God's promises to Abraham and David, thereby declaring that Jesus is the Christ (Messiah).

In a lecture before the Biblical Archaeology Society, Anthropologist Dr. Richard Rohrbaugh explains the significance of Geneologies. He explains that this establishes the reputation and credentials for Jesus. I have transcribed some relevant excepts of this lecture below:

Now, The Baltic Culture Continent that you and I live in, anthropologist call a “Guilt Culture”. ...the Mediterranean Culture Continent both today and in antiquity is what anthropologists call an “Honor-Shame Culture”.

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Honor is relatively simple actually to understand. Honor is your standing in the pecking order of the village, together with the public recognition of that. There is no such thing as claiming honor that the village does not recognize. To claim honor that the village does not recognize is to be uppity; brash; a braggart; a fool. Honor is public reputation in the village and everybody in the village knows exactly where you stand in the pecking order. The reason for that is there are two primary ways in which you can get your honor rating or ranking in a village. ...

The overwhelming way in which you get your honor rating is from your birth. It's what anthropologist call “ascribed honor.” It's the honor that you get the day you pop out of the womb. It's the honor that you and every member of your extended family has - male and female - everybody in your family has, has always had, and always will have. That kind of Ascribed Honor means that if you are born in a very high family, you have a high honor ranking. If you are born a low-life, you have a low honor ranking and you're probably going to have a till the day you die. So the overwhelming way in which you get your honor is from the family of your birth. Do you now understand why genealogies are so important in the Bible? Genealogies indicate in writing what a village knows orally. Namely, the family you were born in and hence the on a ranking you have.

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Genealogies are, if you will, a kind of a map for the whole community - describing exactly where in the scheme of things you fit. Among non-literate people (which in antiquity was about ninety-six percent of the population); among non-literate people genealogies would be very short. Only upper class wealthy people have written genealogies. And you understand that the longer the genealogy, the better? Because it means you're from old money not new money. You understand?

...What I find interesting is the genealogy in Luke - It goes all the way back to “son of seth, son of Adam, son of God”. That is, it traces it to the beginning. That's the longest genealogy possible. Do you understand that in honor claim is being made? In fact we know from Roman texts that people in the Roman world who did become newly rich and wanted to move up the social ladder hired genealogists to create fictive genealogies for themselves and there were a few stars in the pantheon of Roman ancestors they all wanted to be associated with. For a fee, you could get that association. Now you have the map that tells everybody where you fit in the pecking order of things and that had an enormous impact on your life.

I’m going to show you some slides in a minute of some ancient text that describe the fact that your honor ranking determine who ate with whom, who could marry whom, who spoke to whom, who listened to whom. It determined who would speak first in a conversation. It determined who would marry whom and who would do business with whom. In fact it determined most of the social patterns of your life. It's therefore critically important that everybody in the village know exactly where you stand - because it provides the road map for how you and I are going to interact with each other.

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Jesus comes from a no-account little village. He's a village – a τέκτων (tektón) he is called in the Greek. We translate it carpenter – it could be a worker in metal, stone, or wood. I don't have time today but to show you just how low on the social scale that really is - It's very near the bottom. People like that don't get up and talk in public. So when Jesus does, It confuses everybody. In the Middle-East they expect somebody born of a great family to be great. You're born of a low family you're going to be no-account. What does not compute in their social compass is somebody born to a low-life family who turns out to be great. How do you explain that?

Well of course what Matthew and Luke do to explain it is they give us these very elaborate birth stories in which they try to tell us that God was somehow unusually involved in this birth. Otherwise there could be no expectation that anybody would listen to Jesus. What do his opponents do in the twentieth chapter of Luke? They say, “Who gives you the authority to speak like this?” Note that they didn't say, “Did you have this authority from your birth?” They know that's not true. “Who gave you this authority?” Their assumption is somebody had to have acquired the honor from somebody who recognized it because all honor has to be publicly recognized.