You might use one of these frameworks if you want to write code that does networking.
For example, if you were going to write a massively multiplayer video game, "setting up a Java program ... to dispatch a thread for each request" probably isn't an option; juggling that many threads is phenomenally complex, and it performs poorly as well. Not to mention the fact that "just spawn a bunch of threads" is missing a bunch of the management tools that Twisted et. al. have, like
twistd, which handles logging, daemonization, startup and shutdown, and so on.
Or if you wanted to write a build automation system, the ability to asynchronously invoke and control subprocesses would be useful. If you spawn a process asynchronously, you can easily kill that process and gracefully deal with its exit. If you spawn it by starting a thread and blocking in that thread you can't stop it easily, since stopping a thread is inherently unsafe.
EventMachine and Twisted can both be used to write client-side programs as well; maybe you're writing a GUI application that isn't web-based, and you want to use the same protocol implementation on the client and the server.
Since you can use asynchronous frameworks in so many different contexts, it's possible that you might want to use it in a web application simply because you have existing library code, written for some other application using your async framework, which you want to use. Or you might want to be able to re-use your web application code in some hypothetical future non-web application. In this case, it's not that much different than using Apache or Tomcat or whatever in terms of functionality, it just gives you a more general, re-usable way to organize your program.