On the most basic level, Wright's "transgression" is the mere act of wishing to take out books from a library in the Jim Crow South. Though Mr. Falk agrees to let him use his library card, Wright must still forge the notes purportedly from him in order to request the specific books he wants to read—at first, those of H.L. Mencken.
The more profound "violation" that underlies this is Wright's desire to enter the intellectual world, the world of books, from which the whites have excluded him. He's thus rebelling against the whole mindset of the segregated South, where African Americans were perceived, and treated, in a harsh, dehumanizing way. The subtext of this racist thinking was, of course, the whites' fear that if black people were permitted to read and to acquire knowledge, they would be in a position to overturn the status quo and bring an end to Jim Crow—which, of course, they were eventually able to do.
It is significant that Wright names Mencken as the writer whose books first attracted him. Mencken was an iconoclast who shattered many of the notions around which Americans had built their self-mythologizing. He was against organized religion, for example. In the antebellum South and in the Jim Crow period of the hundred years after the Civil War, whites had used Christianity to justify slavery and segregation. So, by choosing Mencken to read, Wright was making the deepest kind of "transgression" possible as he entered the world of books and intellectualism that had been denied to him.