In order to chat with other players type messages into the chat field down in the lower left corner of the screen and hit "enter" or "done" on your keyboard to send the message. That will display your message to everyone else in the room with you.
Once you have typed your message, hit Enter or Done on your keyboard to display your message.
There is also an option to send a Whisper Chat, which is a private message to another player. Click on the icon of the player you want to whisper to and select Whisper Chat. The message you type after the player's name (which will appear in the chat field) will be displayed only to that player, even if they aren't in the same room as you.
Click Whisper to send a whisper to another player.