That the distinction, vegetarian/meat-eating, is important for the transition in Republic II, between the healthy city and the city of luxury, is clear. But why is this distinction important for Plato?
There are two reasons, I think, one proximate and the other remote. I'll explain the proximate here and the remote in the next post.
The proximate reason is that, in Plato's mind, the desire to eat meat is the principal non-necessary desire which is not subject to natural limits, requires land for its satisfaction, and therefore leads to expansionism and war between cities.
The demand for animals used for work, or for warm clothing, is limited. Each family will need only an ox or two for that purpose. The lifespan of an ox is about 30 years; this means that each adult male will need only a single ox for his own lifespan of hard, physical work. Once dead the skin of that ox, if converted to leather, would probably make sufficient leather garments for a single man's lifetime. An ox or two won't need much land for pasture.
But the demand for animals for food is not limited, at least not in any similarly obvious way. Even a small family would need several large animals per year for meat. As their tastes become more refined, they'll be satisfied with fewer and fewer parts of the animal. These animals will need far more land for pasture than would be needed for a family in the 'healthy' city--or land would need to be dedicated to raising grain for them.
Even if the citizens of the luxurious city did not have more children than they should (Plato thinks that they will, perhaps because of the proliferation of 'reciters, actors, dancers, producers' in the luxurious city, who will inflame sexual desire), their habits of eating meat alone will require them to look for more land. This, indeed, is what Plato seems especially to emphasize:
Do we need, then, to carve ourselves a slice of our neighbour's territory, if we are going to have enough for pasturage and ploughing (nemein te kai aroun) ? And do they in turn need a slice of our land, if they too give themselves up to the pursuit of unlimited wealth (chrematōn ktēsin apeiron), not confining themselves to necessities (huperbantes ton tōn anagkaiōn horon)? (373d)(By the way, this surely is Plato's introduction of pleonexia, and thus 'love of money'. It was not present at all in the healthy city.)