If the stock is $100 when it vests, you pay income tax on $100 per share, whether you hold or sell. It's exactly the same as if you got paid that money in cash and then decided to buy shares (after paying tax).
If you then hold and it goes above $100 while you held, you pay capital gains tax (short or long-term, depending on how long you hold) when you sell.
Again, exactly the same as would happen if you bought those shares yourself with a cash bonus. Getting shares really is the same as cash, for tax purposes.
I still don't get how there's any way holding it instead of selling it reduces taxes.
If you sell on vest, you pay income tax and 0 capital gains. If you sell after vest, you pay the same amount of income tax, and possibly nonzero capital gains.
Well... if it's worth $X when it vests, you pay income tax on $X. If it's worth $Y when you sell, you pay capital gains on $(Y - X), presuming you held it longer than a year. If it was less than a year, you pay income tax on $(Y - X).
At least, that's my understanding. I'm not a tax accountant, though. This is not financial advice.