This section deals with Socrates' arguments for a third division of the soul. Previously he has argued that there is "reason," the capacity to do high-minded and virtuous activities. There are also the "appetites," the desires and impulses that demand to be fulfilled. But what determines whether one listens to reason or the appetites?
The term "spirit" here is one way of translating the term; when I first read the "Republic," the translation I read used the term "indignation." What the spirit does is react to circumstances and side with either reason or the appetites and so guide a person's behavior.
The question here is ambigious; I assume you mean what arguments Socrates uses to support his idea of the spirited part? This is how I understnad it, because this section only deals with Socrates' arguments for this third division of the soul.
At any rate, he first argues that there are at least two parts to the soul: reason and appetites. He does this by arguing that sometimes a person can feel hungry, but resist drinking. Next, he notes that sometimes one is conflicted whether to listen to desires or to listen to reason. How does one decide whether to listen to reaosn or appetites? There must be a third function in the soul. That third function is the spirited part.
He next argues that because it sometimes sides with the appetites, it can't be an aspect of reason. And then he argues that because we sometimes we act on our nobler natures, it can't be part of the appetites. This further strengthens the idea that it is distinct from either of these.
Thus he argues that there is a third part of the soul. Its function is to ally itself with either appetites or with reason, and so strengthen that part. One acts virtuously because the spirited part, by its strength and force, sides with reason and tells us the right thing to do. And sometimes that spirited part sides with the appetites, and one must act on one's desires.