> Do stock options holders need to pay tax even if they have not added actual money to their bank account?
Yes. The options are converted into shares, which is considered a taxable event by the IRS. Ordinarily this isn't a problem because you can sell some fraction of the stock that you gained to pay for the taxes. But if the company is not public, you have no easy way to sell the shares.
> If I don't have the money in my bank account yet, how can I pay this huge amount of gains tax even?
That's exactly the problem. Many employees at unicorn startups are stuck in "golden handcuffs." If they leave, they oftentimes can't exercise their options due to the taxes, in which case they lose them. This forces them to stay in the hopes that one day the company will go public and they can exercise their options.