Correct–the plugin takes care of installing the code for you.
However, when I look at your page, it looks as though a widget has not been selected via the plugin. Just to double check, have you selected a widget in WordPress using the Pure Chat plugin?
When you open the plugin, you’ll see a screen that looks like: https://plugins.svn.www.ads-software.com/pure-chat/trunk/screenshot-5.png
Pressing the “Select” button for a widget will automatically add the Pure Chat Javascript code to your WordPress site.
Thanks!
Aaron @ Pure Chat