So I'm afraid I might be missing something pretty fundamental here, but I really can't get my head around this - Why? Why would we want to use those JS MVC frameworks, instead of sticking with Rails, Django, PHP and so on?
What do these JS frameworks give us that can't be achieved by the old web frameworks? I read about SPA, and there's nothing I couldn't do there with ASP.NET MVC, right?
I'm really baffled by hearing all the people at work wanting to leave our current framework for these new ones, and it's much more than just for the sake of learning something new. I am totally up for that, and I've always tried playing around with other frameworks to see what I'm missing, but perhaps these new technologies have something really big to offer that I simply cannot see?
Single page applications provide a better experience by having all page transitions be seamless. This means you never see the "page flash" between user actions, in addition to a few other user experience improvements.
Front-end frameworks also generally provide a common way to interface with APIs. So instead of writing an AJAX wrapper for every page in your site, you just say 'This page has this route (path), hooks data with this schema from that API endpoint and presents it with these templates and helpers.' There are many proponents of APIs, because there are many good reason to write you applications from a service standpoint. This talk sums up a lot of the points in favor of APIs. To summarize:
Orchestrating your web offerings as services makes them inherently decoupled. This means they are easily changed out. All the reasons behind strong Object Oriented design principles apply equally to the larger parts of an application. Treat each piece as an independent part, like a car, and the whole platform is more robust and healthy. That way, a defect in the headlights doesn't cause the motor to blow up.
This is very similar to how a SOAP WSDL works, except you have the auto creation tools right out of the box.
Having well defined touch points for each part of your application makes it easier for others to interface with. This may not ever factor into your specific business, but a number of very successful web companies (Google/Yahoo, Amazon AWS) have created very lucrative markets on this principle. In this way, you can have multiple products supported by the same touch points, which cuts a lot of the work out of product development.
As other point out, the front end framework is not a replacement for the backend, server technologies. How could it be? While this may seem like a hindrance ("Great, now we have two products to support!"), it is actually a great boon. Now your front and back ends can be changed and version with much less concern over inadvertently breaking one or the other. As long as you stick to the contract, things will "Just WorkTM".
To answer your additional question in the comment, that is exactly correct. You use a front end framework for handling all the customer interaction and a completely separate back-end technology stack to support it.
I'm forgetting a few good ones...