Think of Angular directives, Ember components, and React components as attempting to solve the problem of web components in your view.
I'm using react right now in a rails application at work. While it hasn't totally replaced my haml views, I definitely sprinkle it in where appropriate. It does a great job of handling dynamic uis and gives a standard, easy to understand structure to my javascript. I think of it is as a next step beyond jQuery.
In my opinion, React forces you to think of your view as a hierarchy. You are forced to think about what components make up your view and what its responsibilities should be. React components also come with some standard methods in regards to the lifecycle of each components and the state of your data as it flows up and down the hierarchy of your components.
I've really enjoyed the experience thus far. I'm not necessarily a fan of putting any sort of html-like structure in my javascripts, but Pete Hunt, one of the core contributors to React, explains his reasonings quite well in many talks you can find on Youtube.
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Also, Khan Academy uses Facebook React in its perseus q&a system. You can view blog posts by two Khan developers about React here: