Your phone should still be connected to internet for web client to work and all the user messages will still reside on Phone only. This service is only for Android & Google Chrome users. Whatsapp said that they will soon launch this service for all browsers and iOS as well.
For activating desktop version of Whatsapp. Go to:-
Download the latest version of Whatsapp from Playstore. Click on Menu>>Whatsapp Web and then scan the QR Code as shown on web browser.
After scanning of QR Code, you will be able to see the messages on your desktop in Google Chrome browser.
Let us know your views about Whatsapp desktop version in comments section below.