I'm trying to use Javascript to take the separate address components from the Google Address field and store them as custom fields. For example I would be able to get the City, Postal Code, State, Country, etc.
I watched your video, and see that you are doing this with a front-end form (the original thread was aimed at back-end post edit screens), and you seem to be most of the way there, inasmuch as you have populated fields in that front-end form with details from the geocoding result.
If those values don't seem to have been saved (when you check the back end) after submitting the form, it sounds like you maybe added generic fields to your front-end form rather than the custom fields themselves (you must have registered some custom fields if I can see them in the back end in the video). In that case, the generic fields would be discarded after submitting the form.