Matthew never calls himself the writer in the book but the early church uniformly attributed this gospel to Matthew. The Didache, the Epistle of Barnabas and the writings of the church fathers, Ignatius, Papias,  Irenaeus, Origin and Eusebius, all say that Matthew was the writer.

For example, Origin writes: “Among the four Gospels, which are the only indisputable ones in the Church of God under heaven, I have learned by tradition that the first was written by Matthew, who was once a publican, but afterwards an apostle of Jesus Christ, and it was prepared for the converts from Judaism.” So the book was known early and quickly accepted.