Schwartz is ‘black’. I’ve heard that an Egg is a mountain ridge. So Schwartzenegg would be ‘black mountain ridge’. A Schwartzenegger would be a person from Schwartzenegg, much as a person from Berlin is a Berliner.
Egge is, I think, the name of some sort of a garden implement. So Schwartzenegger might be a dark (haired, or perhaps complected) person who uses the implement. But I think ‘a person from Schwartzenegg’ makes more sense.
Schwarzen is definitely ‘black’ and egger almost certainly derives from ‘egge’ meaning harrow or a tool like a plow. However, my understanding, which is confirmed after consulting with a resident German speaker here, the name actually means a “plower of black fields” as opposed to a “black plower of fields” .
My best guess is that it means “man from Schwarzenegg” - there is one Schwarzenegg in Switzerland but a number of municipialities called Egg in the German-speaking areas, so Schwarzenegg looks like a likly toponym.
The problem is that his name today has gone through different variations and phonic changes, possibly even spelling changes*, so only somebody who delves into the family history of this family of the Schwarzenegger can talk with any certainty whether it’s a man from a town called Schwarzenegg (related to a black mountain ridge) or a black(haired) plower/plower of black fields.
Unless Arnold has done family history research himself, his own explanation is no more valid than the above guessings.
One example: a current name would be translated as related to hammer, but is really a phonetic corruption, related to the family moving twice, of heaven.
Another common surname can relate to a balance or a waggoner (waggon-maker) or somebody who makes wheels, or something else.