In order to chat with other players type messages into the chat field down in the lower left 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 you keyboard to display your message.
If you have verified your security Email, you can also Whisper Chat. 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.