The Gospel of Matthew was likely written between 80 and 100 AD. Until recent times, the authorship has been attributed to the apostle Matthew; but the Gospel never explicitly states who the author is. The primary argument against Matthew’s authorship is that he uses a lot of material from John Mark, who was a Gentile, and if the book were written by a Jewish author, that wouldn’t have happened.
No matter who the author is, the audience is fairly clear. It was most likely written to a Jewish audience. There is an emphasis on the Jewish Law, traditions, and Jesus’s interactions with the Pharisees and Sadducees. Much of the book consists of quotes from the Torah and the prophecies that Jesus fulfilled.
One of the biggest retorts that the Jewish leaders of the day had against Jesus is that He didn’t save them from Roman rule, because they thought they were getting a political savior. Jesus puts a lot of emphasis on Jesus saving people from their sins and His love for the least instead of the greatest, which would have been a big deal in “proving” that He was the Messiah and Savior in some way.
Overall, the purpose of the Gospel of Matthew is to prove that Jesus is Messiah and that Christ has authority over all of the nations. Because of that, many churches have decided to utilize this Scripture in a number of ways, including the telling of the Christmas story, sharing how Christ would work among people, and to show how much of a difference we can make in the world that we live in.