In the Gospel of Matthew, there is specific theme of "fulfillment" where the author uses OT texts to prove Jesus is the Messiah. This is know has the "fulfillment formulae" of Matthew. The formulae teaches us two things about the Lord. First, God's saving promises in the OT are being fulfilled. Second, God's saving promises in the OT are being fulfilled in the person of Jesus Christ. Schreiner points out how "the fulfillment theme in Matthew demonstrates that OT prophecy anticipated and pointed towards Jesus." The various Scriptures that Matthew quotes teach us that Jesus is the new King (Matt. 2:1-6), the true Israel (Matt. 2:13-15), the new exodus (Matt. 2:16-18; 4:1-11), and the new covenant (Matt.4:14-16). Jesus ultimately fulfills the Scriptures at his baptism when the Holy Spirit descends upon him to "fulfill all righteousness" and identify him as the Servant of the Lord in the OT who came to suffer and die for the sins of his own people (cf. Matt. 3:15; 8:14-17; 12:18-21; 21:5-6; 26:50-56).
The emphasis on fulfillment in Matthew is important to understand because the author was writing to Jewish Christians who were still trying to live in old law of Judaism and not in the new law of Christ. The defining words by Jesus in the Sermon on the Mount is the most profound fulfillment formulae usage that is not quoting an OT text to substantiate the claim, "Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them. For truly, I say to you, until heaven and earth pass away, not an iota, not a dot, will pass from the Law until all is accomplished" (Matt. 5:17-18). Even thought these two verse to not mention an OT text, they do encapsulate the idea that Jesus came to fulfill the Scriptures because they were always intended for him to accomplish and bring salvation to his people because man could never fulfill and accomplish the law.
The fulfillment formulae reveals how God's saving plans are realized in Jesus Christ. Thus, Matthew's Gospel ranges from Abraham to David, to all of the Law and the Prophets, to showcase how Jesus' birth-life-death-resurrection was not an accident, but was the promised-plan from the beginning. Indeed, the fulfillment formulae displays the inauguration of God's saving purposes that was proclaimed and prophesied from the beginning in order to confirm his messiahship who will consummate all things in the end.