In Republic I, Thrasymachos introduces the idea that when we are speaking precisely, the practitioner of a craft, insofar as he is a true craftsman, never errs. This idea of the craftsman as such is familiar from Gorgias, but is more starkly developed here. Thrasymachos needs this distinction to save his definition of justice ("the advantage of the stronger") from the obvious fact that those in power often order, by accident, what is actually to their disadvantage. Polemarchos and Cleitophon had suggested that he change his statement to say that justice is what the stronger thinks is to his advantage, but he rejects this escape route (it would be interesting to imagine what the consequences of taking this alternative might have been).
His position is a standard sophist's relativist/conventionalist one, holding that it is only by custom (not by nature, which he conceives inHobbesian terms) that we call justice a good thing; it is the moral rhetoric of the weak (what Nietzsche later called "slave morality"). I suspect this critique of conventional moral language is the core of why he thinks he can reject Socrates' view, even though he can't defeat his argument -- the pull of conventional usage is just too strong, he might protest -- plus, if he leaves against the will of the young men present, he loses face with them (and thus the chance to command their fees as a sophist). By staying and acquiesing to Socrates, he holds out the promise of this brave new worldview, to which the young men are clearly drawn (they don't want him to leave, and Glaucon and Adeimantos later pose their thought experiment, in Book II, in defense of it).