Daniel von Fange wonders about continuation-based web serving. He adds "treating a 'page' as a method of an object makes more sense to me for 97% of things I do.".
Without getting into how continuations work, you can look at it this way: continuations allow a web page to behave just as you prefer, kind of like a method of an object.
What if methods in your favorite language required you to do the following: every time the method itself calls another method, the return value does not go back to the caller. Instead what if you had to name some other method to receive the return value?
In this case you have to "name the continuation", that is, provide an explicit place (method or page) for the computation to "continue" following the call (or the user interation, in the web scenario).
But in a continuation-based web server, just as in most programming languages, the system handles "where to go next" implicitly. That is, in most programming, you do not need to provide an explicit "continuation". The system is happy to return the value right back to the point of the call.
And so just as your method calls another method and the results are returned right there for you to use, your web page "calls the user" and the results are returned right there on the page for you to use. You do not have to name another page to receive the results.
Hope this helps. The mechanics are less important for understanding the benefits.
If you think about it, "continuation-based web servers" should be the expected behavior since they handle continuations for you, just like programming languages. More typical web servers should be considere the odd ball. They make you do explicit "continuation passing". No one likes to program in "continuation passing style". It's just, we've gotten used to the burden with most web servers.