I've written a lot of posts over the years arguing for the apostle Matthew's authorship of the first gospel. I want to put together a collection of links to several of the arguments, so that they can easily be accessed in one post. I'll add more links as they become available. What I'm linking are posts that address the relevant issues. Since some of these posts address more than one subject, you may have to search within a post to find the material you're looking for.
The apostle Matthew is an unlikely candidate for a mistaken or fabricated authorship attribution.
It's unlikely that the author would have been universally unknown in our extant sources if he was somebody other than Matthew.
An author would have been named by means of titles, tags, and labels attached to the early manuscripts.
The gospel of Matthew is highly Jewish.
The gospel of Matthew addresses the apostle Matthew's conversion differently than the other gospels do.
The objection that the gospel should say more about Matthew's conversion if it was written by Matthew is insignificant.
There's evidence that the author had more knowledge of and interest in money than the average person. (Matthew was a tax collector.)
The objection that the apostle Matthew wouldn't use so much of Mark's material doesn't carry a lot of weight. (See here also.)
It was common for authors to refer to themselves in the third person, so the third-person references to Matthew in the gospel aren't a significant objection to Matthean authorship.
The early prominence of Matthew's gospel is best explained by Matthean authorship.
The gospel of Matthew's having more prominence than the gospel of John is best explained if Matthew was authored by an apostle and was written significantly earlier than John, which places the gospel of Matthew well within the apostle Matthew's potential lifetime. (See the 7:29 P.M. post on 9/5/13 in the comments section of the thread.)
The similarities between Matthew and the other Synoptics make more sense if Matthew was written within years of the other Synoptics rather than merely within decades. The evidence suggests that the other Synoptics were written no later than the mid sixties.
What Matthew does and doesn't say about eschatological issues and the destruction of the temple is best explained if the gospel was written prior to the destruction of the temple in 70 A.D. (See my post in the comments section of the thread.)
Early Christian sources referred to the author as Matthew.
Early non-Christian sources referred to the author as Matthew.
The attribution to Matthew was widespread and persistent, and the earliest objections to it were late and weak.
The attribution to Matthew occurred at a time when false authorship attribution was widely condemned and the authorship attributions of other documents were disputed far more than Matthew's was.
The earliest sources who made authorship judgments had access to a lot of significant evidence. They probably made use of much of that evidence in the process of reaching their conclusions about authorship issues.
The tendency of so many scholars and other individuals to be so unconcerned about external evidence for authorship, while relying so much on such speculative arguments about internal evidence, is a recent development that doesn't make sense. (See here as well.)