How do different blockchains communicate? I've learned that the different cryptocurrencies have their own P2P blockchain networks. 

How do exchanges convert coins from one network into a coin on another network?
Cryptocurrency exchanges are generally taking custody of your funds while you are using their service. You send a deposit of fiat to their bank account or cryptocurrency to an address under their control and after this payment has been confirmed, they credit you a balance on their platform. When you trade one currency for another, they only update their liabilities to you in their database. Only when you withdraw, they send you the corresponding cryptocurrency to your address or wire you fiat.

As you see, the exchanges simply accounts for what they owe you and there is no need for various blockchains to communicate beyond the exchange sending and receiving payments in various currencies.