What is the main argument in Plato's Republic, Book VII?
A key element of Plato's Republic is that it is written in dialogue form. Rather than having a singular "main argument", the dialogues explore all sides of a question, often showing how people's reasoning about a subject is grounded in their personalities and habits of thought as well as reason. The dialogues were probably used in Plato's Academy as starting points for discussion and investigation rather than as providing final answers to questions. In a sense, looking for a conclusive "main argument" in Plato's dialogues is to misunderstand the nature of the dialogue as a philosophical genre.
The key to following the discussion in Book VII is to remember that the dialogue is a struggle to understand the nature of the soul. The participants agree that one way to study the soul is to study the polis or city, as the city is, in a certain way, the soul writ large.
One of the most philosophically important parts of Book VII is the allegory of the cave. In this, humans are compared to people living in a dark cave and only experiencing the outside world as two-dimension shadows. The shadows are the equivalent of what Plato terms the phenomena, things as we perceive them are as they appear (these are sometimes called "appearances") Those shadows are not the actual reality of the outside world. The philosopher strives to leave the cave and see things as they actually are, or the Forms (sometimes called "Ideas" or "noumena") which cast those shadows. Socrates likens the philosopher or educated person to someone who has stepped outside the cave and tries to explain to the people who have always lived in the cave about the nature of reality outside; such a philosopher's ideas may be rejected, considered strange or even insane, by people whose world is circumscribed and who are attached to their own ignorance.
The rest of Book VII focuses on an ideal education, with a strong emphasis on mathematics and logic. The main concern of the interlocutors is that rulers should have true knowledge and strong reasoning abilities, rather than only being aware of the superficial images one sees from inside the cave.