How then does the purchase take place and how does the purchase and sale of tokens on binance affect token prices and how?
Binance holds some amount of each token traded on their site, and manages the user balances and trades in their own orderbook and databases.
Only when someone withdraws the tokens from the site, Binance performs the withdrawal from one of their addresses, which is recorded on the blockchain. Same goes the other way around for token deposits.
Note: This post is valid for the widely-used Binance (centralized) exchange. There's also Binance DEX that uses the Binance Chain (predecessor of the currently widely used Binance Smart Chain) to manage all orders and trades.