REST as "representational state transfer" is a boolshit. The real reason it is popular - it allows to communicate via HTTP, just URLs everyone understands and many tools available, even web browser can be used. If you remember SOAP - the crazy approach to web services promoted before REST, you understand the difference REST made. The same way AJAX is not about XML, REST for me is not about "representational state tra…
Feel free to expand those explanations... :)
The REST culture gives us some regularity. If we read github API, or google API or facebook API, we know what to expect - that they will use GET to retrieve an object, DELETE to delete an object and so on. It's good there is a convention helping to study new things.
Another good property of REST culture - it prevents people from building thick layers on top of HTTP. While not everything can fit into HTTP (at least error reporting, in my opinion; HTTP error codes are not enough), this culture helps to keep APIs simple.