The conjectural question of why tragedy is central to Steinbeck's book is quite different than the formal question of how tragedy is central to the work.
To begin, we might clarify that tragedy of (or in) characterization is a somewhat loose usage of the idea of tragedy. It makes sense, but may be taken different ways. (We might presume that that characters are associated with ideas of fate or inevitable demise, which I believe is the idea here...and the question is why is this so?)
To answer the first question we will have to consider the nature of the text and Steinbeck's intentions with it. The work was written during the Great Depression, a time of serious difficulties for many Americans and especially for the poor. The novel clearly directs its focus to labor issues concerning 1) the poor and 2) the conflict between capital and labor. Steinbeck was a political writer, in large part, and his politics are on display in this work. One statement the novel makes is that the American Dream of land- and home-ownership is actually an impossible dream, or better yet a fantasy, for migrant workers like George, Lennie, and Candy.
"Someday—we’re gonna get the jack together and we’re gonna have a little house and a couple of acres an’ a cow and some pigs and—.”
This dream will never come true. There is an idea at the heart of this book that these men are driven to dream of a better life that is denied to them by a capitalist system at the time that was organized to serve capital and to keep the laborer from getting "ahead".
Why are the men given a "tragic characterization"? Their plight as part of the class of the working poor was fundamentally "tragic" in the sense that a simple hope, for George and Lennie, is so unlikely as to be best seen as a fantasy. The characters of George and Lennie are closely associated with the notion of a negative fate as their combined end is suggested in the opening chapter of the book when George makes Lennie memorize an escape plan.
The story itself is more likely to be discussed as a tragedy than are the characters to be described as being "tragically characterized".
In literature, tragedy refers to any composition with a somber theme carried to a disastrous conclusion. (eNotes)
The political themes are certainly somber in this work, as are the narrative elements and more humanistic themes (of isolation, violence, and negative fate). The novel's conclusion is disastrous.