Building a stateful web application is possible, but not feasible for most application scenarios. As for why a person may want such a thing? LOTS of reasons. An obvious example is a browser-based chat application. You want all members of the chat to see all messages entered as close to real-time as possible.
The most common and easy to implement solution is to build an "AJAX" solution. If you don't know what AJAX is, search here in the forums or just Google it. In short, it is a way for javascript running in the browser to make separate HTTP/HTTPS requests to your web server. It can do this "in the background" (asynchronously) while the user continues to use the browser application.
So while not truly stateful, you can build a web app that appears to be stateful by having an AJAX request check frequently for server-side changes and then update the client browser accordingly. This could mean that the AJAX request actually retrieves the new information and dynamically changes the page contents or it could mean you fire a page refresh.
Depending on the number of simultaneous users you need to support and the power of your server hardware and bandwith, you can set your AJAX request to fire every 5 seconds or whatever makes sense for your app. You may be able to get away with 3 seconds, but 1 or 2 is simply asking a lot of a browser/webserver relationship. If you code it correctly, …