"...regarding his son, who as to his human nature was a descendant of David..." (Romans 1:3)
The gospel was promised by God through the Holy Scriptures regarding His son Jesus. Paul is starting to build his case and it is not by chance that the starts with the fact that Jesus was a descendant of David, by his human nature. He does this to engage the Jewish reader and to support the claim that in fact the Good News of the Messiah has come. The Jews wouldn't be able to receive any of this information without first recognizing that Jesus' blood line was one and the same as the promised Messiah.
This is also a very important piece of information to us as gentiles because when we look at the full spectrum of the scriptures both old and new it is important for us to compare all the evidence of Jesus against the prophecies of the promised Messiah. Paul knows this is a prophecy of the scriptures and on that needs to come to pass if Jesus was in fact the Messiah or Christ.
One obvious place to look for the prophecy is Isaiah 9; if you grew up in a church or merely went to services on Christmas then this scripture should be familiar to you.
6 For to us a child is born,
to us a son is given,
and the government will be on his shoulders.
And he will be called
Wonderful Counselor, Mighty God,
Everlasting Father, Prince of Peace.
7 Of the increase of his government and peace
there will be no end.
He will reign on David's throne
and over his kingdom,
establishing and upholding it
with justice and righteousness
from that time on and forever.
The zeal of the LORD Almighty
will accomplish this.
( Isaiah 9:6-7 )
This is an obvious prophecy of the Messiah and one which foretells that the Christ will reign on David's throne. Now we are well aware of the fact that to reign on the throne of a king one must be a blood descendant.
The height of Israel's royal kingdom was during the reign of King David and King Solomon and Isaiah said that the Messiah would ‘reign on David's throne and over all his kingdom, establishing and upholding it with Justice and righteousness from that time on and forever.' So to be the Messiah who could accomplish this it would be important that the Christ was of David actual blood line.
God also promised David that his kingdom would be established forever. II Samuel 7:16 says, ‘Your house and your kingdom will endure forever before me; your throne will be established forever.' This was all part of God's promise to David and this promise is also recorded in II Chronicles 17:14, "I will set him over my house and my kingdom forever; his throne will be established forever.' And lastly it is David who wrote about this promise from God in Psalm 132, "Here I will make a horn grow for David and set up a lamp for my anointed one." ( Psalm 132: 17 ) Here the word horn symbolizes strong one, that is king... and the ‘anointed one' is common as a descriptor for the Christ throughout the Old Testament.
So throughout the Old Covenant it is apparent that the Messiah would come through the bloodline of King David. And if you flip over to the beginning of the New Testament the written record of Matthew's gospel begins with the genealogy of Jesus...
"A record of the genealogy of Jesus Christ the son of David, the son of Abraham..." ( Matthew 1:1 ) As you read through the genealogy you will see that this genealogy records Jesus' ancestry through Jesus' earthly father Joseph and that ancestry was traced through the royal line of David and King Solomon. "And Jacob the father of Joseph, the husband of Mary, of whom was born Jesus, who is called Christ."
Matthew clearly writes to a Jewish audience, so the lineage traced through the father confirming the royal bloodline of King David would be important to them when deciding if Jesus was the Messiah.
Now, if you look at Luke's account of the genealogy of Jesus, you are going to see a completely different genealogy of Jesus, you are going to see a completely different genealogy. But his ancestry is also traced back to King David, but is trace back through one of David's other sons, Nathan. Luke who wrote this gospel was not Jewish so tracing the lineage back to King David through Joseph the earthly father wouldn't have been important to him. What would have been important to him was to record accurately the account of Jesus, which was the declaration at the beginning of His letter, "Therefore, since I myself have carefully investigated everything from the beginning, it seemed good also to me to write an orderly account for you, most excellent Theophilius, so that you may know the certainty of the things you have been taught." ( Luke 1:3-4 )
Luke's focus and purpose was to record accurately the account of Jesus' ancestry to us, who are not Jewish, so we could see and understand how Jesus descended from David thus fulfilling the Messianic prophecies concerning the Christ coming from the line of David.
Luck gives us one other important piece of evidence here in his writings... Luke 3:23 says, "now Jesus himself was about thirty years old when he began his ministry. He was the son, so it was thought, of Joseph." First he gives this little disclaimer that it was thought that Jesus was the son of Joseph and then launches into a completely different genealogy then that of Matthews account.
So at first glance, this seems to contradict itself, but upon further study and reflection we see that Luke's attention to detail actually adds to the fullness of the fulfilled prophecy because Luke actually gives us Jesus' ancestry through his Mother Mary, which her ancestry can also be traced back to King David and since Jesus was not of his earthly father's bloodline it was necessary to trace his bloodline to King David through his mother Mary.
So Paul knows the importance of his for both the Jews and the gentiles that is why he starts out the book of Romans assuring his readers that Jesus was the promised ‘Good News' foretold about in the scriptures through the prophets and in accordance to His human nature, he was a descendant of David, thus fulfilling scripture and prophecies about the Christ or Messiah.