Let’s work through it with numbers, in case that isn’t clear. Say Saverin has $100 worth of Facebook stock. He renounces US citizenship. He sells enough stock to pay $15 to the government, leaving him with $85 in Facebook stock. Over the next year, Facebook doubles in value, leaving him with $170 in stock. He cashes out, and there’s no tax, so he ends up with $170 in cash.
Now imagine he stays in the US. He has $100 in Facebook stock, which he pays no tax on because there is no taxable event. Facebook doubles in value over the next year, so he ends up with $200 in stock. Then he cashes out, paying 15% in capital gains, leaving him with $170 cash.
Clearly he ends up with the same amount of cash either way. Perhaps Saverin is avoiding tax by leaving the US, but neither this article nor any other article I have read identifies how he is doing so.