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I am still unclear after reading the article.

What is the exact interface where a software change leads to a change in some physical word state?

For instance, logically storing X in some register is going to lead to some particular configuration of electrical current on the CPU. I still don't see the exact point where that happens.

How do the two intersect?

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I highly recommend the book, and the course, http://www.nand2tetris.org/ (previously known as The Elements of Computing Systems or TECS).

This starts with discussion on NAND gates and goes through building CPU components, to a complete computer, to implementing a high-level language for it.

(All the chapters from the book revevant to your question are available as free PDFs).

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