In Matthew 1:1, it says that Jesus is the son of David and the son of Abraham. Why does he not start from Adam and Eve?
This question spawns from a misunderstanding of why the genealogy is there in the first place.
The genealogy is not a benign collection of facts, much like our own western approach to the topic would have it listed. Instead, Matthew provides Jesus' genealogy to legitimize him for any Jewish readers.
First, consider that everyone is from Adam. It means nothing for any purpose to start a genealogy from Adam. We all came from Adam. If Matthew was simply stating the facts of Jesus' family line, then starting at Adam would have been acceptable. But Jesus needed to be legitimized for a Jewish audience.
Abraham is the father of all Judaism. That's not just a colloquialism. Abraham literally is the ancestor of all Jews, by both blood and marriage. The genealogy starts at Abraham to tell the audience that Jesus was indeed a Jew.
Judah, the son of Jacob, stands out as the forefather one of the twelve tribes. This tells the reader that Matthew is claiming Jesus is from the tribe of Judah.
David stands out, and is probably the most important, because of specific messianic prophecies that the Messiah must be from the line of David. Matthew claims Jesus is the Messiah, so this one name in Jesus' genealogy is vitally important.
In contrast to modern, western thinking, Matthew used Jesus' genealogy like an address, proving that he is the Messiah. There is a very obvious appeal to Jewish readers. A number of other things stand out, such as the fact that it is listed in three sets of fourteen, indicating a kind of poetry, and the heavy annotations in the first set, but missing from the other two.
Compared to Luke's genealogy, which is more utilitarian and fact centered, tracing from Jesus to Adam, it is clear that Matthew's genealogy serves a very specific purpose.
The sources on this are abundant across the internet. The Wikipedia article Genealogy of Jesus is a good start.
Modern scholarship tends to see the genealogies of Jesus as theological constructs rather than factual history. Thus the two New Testament genealogies should be understood in terms of what they were meant to achieve, rather than as a collection of facts.
Matthew and Luke provide detailed genealogies for Jesus, back through the great Zorobabel to the line of King David. The author of Matthew wanted Jesus to be descended from the royal line throughout, whereas the author of Luke wanted Jesus' ancestry to remain commoners back as far as David. Thus, in Matthew, the paternal grandfather of Zorobabel is Jeconiah, whereas in Luke he is a commoner called Neri.
Luke's Gospel has great men occur in multiples of 7 generations starting at Adam (inserting Kainan at the 13th generation to achieve this), with Abraham at generation 21 and David at generation 35, with Jesus at 77 proving he too was destined for greatness.
Matthew's Gospel demonstrated (Matthew 1:17) that there were 14 generations from Abraham to David; from David to Josiah (omitting three kings in the Old Testament list); from Josiah to Jesus, so also proving Jesus was destined for greatness. However, the Old Testament only lists 20 generations from Adam to Abraham, whereas this genealogy would require 28 generations from Adam to Abraham to work as intended. Thus, Matthew's genealogy does not go back to Adam, but only as far as Abraham.