Division by 7, the Moura Velho Method

This week, Silvio Moura Velho showed me a new method for determining if a number is divisible by 7 (and by 13).  The method is described at: http://www.7and13divisibility.com/. I am impressed at how quickly the method works.  What I feel is missing on the site is a good explanation as to why it works.  (The "Why it works" tab seems to only provide another example.)  Can you explain succinctly why this method works?

Source: Original.

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