The demand that the characters be consistent is in many ways parallel to Aristotle's demand for the unity of plot. Every action in the plot should be causally connected to every other action. The tragedy, viewed as a whole, should have the internal consistency of a clock, so that we should see a near inevitability in the way things turned out. Similarly, an agent should behave in such a way that every decision, every action, can be read as a manifestation of a single, unified character. Characters, too, should have the regularity of a clock, so that, when viewed as a whole, there should be a seeming inevitability in every decision the hero makes, based on what we know of the hero's character.
Aristotle does not rule out entirely that a hero could behave inconsistently, but he demands that the play, seen as a whole, should make this inconsistency comprehensible. Though in one instance the hero may behave one way, and in another behave in a contradictory manner, this contradiction should be made clear by the larger context. Aristotle condemns plays where inconsistent or puzzling behavior is never clarified.
Aristotle's hero must be of high rank, relatively virtuous, true to life, and consistent. These requirements depend to some extent on a relatively transparent moral worldview and understanding of psychology. In a world where motives are unclear and there are layers of psychology to work through, it might be difficult to determine whether a character is ultimately "good," or what goodness consists of. Further, a character may seem inconsistent, or at least ambiguous, if the agent's motives don't float to the surface by the end of the play. Euripides in particular is known for writing plays full of moral and psychological ambiguity. Not surprisingly, Aristotle seems to prefer the much cleaner Sophocles to Euripides. In retrospect, though, this seems more a matter of taste than of irrefutable reasoning.