Game chat is basically like a phone. You can talk with whoever you're trading/battling with for easier communication.
In order to use it, you must have it enabled. The person you want to chat with must also have it enabled. However, you can only game chat with people on your friend's list. When you want to trade/battle, it'll ask you if you want to game chat with them. If the person agrees, you'll be able to talk to them while battling or trading.
Hope I helped!