I have been reading about various Flex frame works, Robot Legs, Swiz, Cairgnom (spelling error) and a few others. What I am missing is how they actually work with Flex / Flash Builder. Are they simply libraries that you can use with the Flash Builder IDE? Why should I learn to use one of them instead of just using the IDE as it is?
We use Cairngorm in our projects as follow:
Like this, we are able to use Cairngorm classes
You can do the same thing with Astra Flex and use graphical components.
They essentially try to provide basic functionality that is (hopefully) useful across many projects. Most of them provide an implementation of MVC patterns. The idea is that you don't need to spend your valuable time writing "plumbing" code, as the framework has done that work for you.
These frameworks often demand a large up-front learning cost to figure out "their way of doing things", but have the potential to save you time over the long run as you re-use the framework & knowledge over multiple projects.
Destroyica's answer details how exactly to wire these frameworks up. Most provide a single .swc file that you link into your projects' libs\ folder, though some will offer you the source code to the project that you could add to your projects' src\ folder.