Having internal mental experiences causes my brain to send physical signals to my fingers in order to type the words "I have internal mental experiences". A philosophical zombie would type those same words, but they would be caused by something else since by definition it lacks those experiences. That would be rather surprising, and it would be even more surprising that the words that the zombie emits coincidentally correspond exactly to the experiences that the non-zombie has.
As the article says the alternatives (that the author seems to favor instead) boil down to "there's some physics we missed" and probably that's the point where we differ. I find that implausible, what would that be? Consciousness quantum field? Consciousness boson? If it's going to be interacting with matter it has to have a way to do that.
Internal experience is an input to your mind (that’s how you know about it), and what your mind perceives affects what it does. The better question is: how can what you experience possibly not affect your actions?
The need arises when you consider the complex social systems humans leverage for survival — our ability to navigate interpersonal conflicts and engage in cooperation relies on theory of mind and our tendency to perceive ourselves as individual free agents empathizing with separate free agents.
By definition, internal experience is something you perceive, and what you perceive informs how you act. Therefore internal experience affects outward physical action.
(See https://www.lesswrong.com/posts/fdEWWr8St59bXLbQr/zombies-zo... for a much longer version of this argument)