Plato argues that justice is intrinsically beneficial to the soul, even in the absence of any external advantages. Plato's conception of justice is eudaimonistic because maintains that justice benefits the soul. This claim stems from an ethical theory that should be distinguished from two other common ways of thinking about ethics.
On the one hand, eudaimonism is distinct from egoism insofar as the two theories disagree about what truly is advantageous. For the eudaimonist, a thing is ethically good if it benefits the soul. Additionally, the soul is often thought to be social, so eudaimonistic goods often involve concern for others. For the egoist, something is good if it benefits the self. In this view, someone is acting well when he does what is to his own advantage, regardless of concern for others. Hence one of the main differences between the two theories is the extent to which one is communitarian and the other is individualistic.
On the other hand, eudaimonism differs from deontology insofar as the former concerns advantage and the latter excludes it. The eudaimonist thinks that something must benefit the soul in order to be good. The deontologist regards advantage as a mere contingency, and bases moral judgments on their adherence to universal principles. For the deontologist, an action is good, even if it makes one worse off, if it is the sort of action which is good. Deontological theories have different criterion for determining what sort of action is good, but most include universality and non-contradiction as standards. Thus eudaimonism and deontology are distinct in that one considers advantage and the other does not.
Plato is a eudaimonist because he argues that justice benefits the soul. Thrasymachus is an egoist who thinks that everyone should do what is to their own advantage, without regard for the disadvantage of others. In the Republic, the modern deontology is suspiciously missing. Hence one should not expect to find claims that something is good in and of itself, but rather every ethical claim will refer to some benefit for some party.