What is the significance of the title of the Book of Matthew? Is the title a metaphor or representative of a specific incident in the book?

The first issue you should consider in addressing this assignment is that the Book of Matthew, as is the case with many ancient works, may not have had a title in the modern sense. Thus, talking about an "author" choosing a "title" fundamentally misunderstands the nature and history of this text.
The Gospels (Matthew is one of the four Books of the Gospel) which eventually were included in the New Testament originated in oral tradition as various stories concerning the life of Jesus and collections of his sayings. This particular work synthesizes materials from the earlier Gospel of Mark with a sayings source referred to by scholars as "Q". 
The title in Greek is "To kata Matthaion euangélion," literally "the gospel [good news] according to Matthew." The translators of the Authorized Version (popularly known as the King James Version) translated the Greek into English quite literally.
The title suggests the work tells the story of the "good news" brought by Jesus following a tradition originating from Matthew, one of the twelve disciples of Jesus. Given the reliance of the text on Mark and the fact that the earliest attributions of it to Matthew occur in the second century, it is unlikely this was actually an account written by Matthew, although he may have been a source of some of the oral traditions that found their way into the text. 
The reason for the attribution of the work to Matthew arose in the context of second century debates about and contests for authority among very early Christian factions. Irenaeus and Papias were especially concerned with establishing apostolic succession as a basis for doctrinal authority, and thus the significance of imputing authorship to Matthew has to do with using a form of what rhetoricians of this period would have termed extrinsic ethos as grounds for establishing a text or doctrine as authoritative.