I do not think that the explanation conveys what the author intended. I think they were thrilled with ExtJs AJAX APIs that are easy to operate and can be invoked using several GUI components (what they mention).
The AJAX API and the API component do not create a server-side web service API for you, which, in their opinion, leads you to the idea:
This gives us the advantage of exposing the same AJAX (JSON) calls as web services APIs to our customers ... Using Ext JS, we have reduced the cost of developing web services APIs.
Well, not really, ExtJs does not create a web services API for you; it creates a method for interacting with your custom REST (or any other) API. Although they might have in mind that, since ExtJs defines how the client should interact with the server using their many components, their server services are easier to create. For example, ExtJs simplifies working with REST web services and defines expectations in callbacks, etc. Thus, in ExtJs, which provides a client solution, it was easier to develop a server solution (web services API) with half the contract.
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