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These two benefits can also apply to web APIs. If our resource objects contain hypermedia, then our APIs become self-documenting. If the system that communicates with our API understands that it will receive hypermedia then we will achieve client/server independence. Our APIs, then, can evolve much easier (as well as the client relying on our APIs). This Martin Nally article doesn't discuss REST or HATEOAS directly, but it does provide a very thoughtful and compelling reason for using hypermedia (aka links) in APIs.
Engine
In this context, engine is an abstract term meaning the force that produces a result. By itself it doesn't mean anything but it does connect the force (hypermedia) with the result (application state).
