Prices set by firms are the same in a Salop circle. But why?
tl;dr: Prices are identical because firms are identical.

Profits of firm $i$ depend on own price $p_i$ and on neighbors' prices $p_{i-1}$ and $p_{i+1}$. Prices are set simultaneously, therefore competition in this market is described by a (normal form) game. A strategy is a price, and your formula for $p_i$ is the best response function, showing firm $i$'s profit maximizing price, given its neighbors' prices. Since consumers are distributed uniformly along the circle and the firms are equidistant and have the same marginal costs, the game is symmetric. It is therefore natural to look for a symmetric Nash equilibrium as a solution of the game. To do this, you assume that your own price and your neighbors' prices are symmetric, i.e., identical, while at the same time your price is the best response to their prices. This gives you the common price of all $N$ firms in symmetric Nash equilibrium.