These frameworks are designed to make your code easier to maintain and to improve the user experience by creating single page applications, in this kind of apps the navigation is more fluid because everything is loaded via AJAX if you want to learn more about it you can start with http://en.wikipedia.org/wiki/Single-page_application.
Another advantage of this sort of applications is that you are going to need an HTTP API to be the access point to your data access layer (DAL) from the front end. When you have a HTTP API your DAL is 100% independent of the data presentation layer this means that in the future you will be able to re-use it to create other apps like for example mobile apps, you can even use tools like phone gap to auto generate the mobile apps for you if you follow this approach.
You can also save a really big amount of time auto-generating your HTTP API by using tools like http://deployd.com/. So we can say that following this approach provides a better user experience, is faster to develop and easier to maintain and test that's why it is so popular. Big companies like Google (angular.js) and twitter (https://github.com/twitter/flight) are using it but they just created their own one.
If you find complicated to decide what framework to use take a look to http://todomvc.com/ they provide with the same example (a TODO application) in each of the available frameworks so you can pick the one that you like more.
Hope it helps :)
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