Options are underwater if the strike price is less than the current market price; shares in private companies might not have an actual market price but you can estimate them. Underwater options still have >0 value since there's a non-zero chance the stock price will decrease before the expiration date -- of course, if the expiration date is close by and the strike price is much higher than the current price, the option value might be very close to 0. Still, you practically never exercise underwater options (there are some rare exceptions not very relevant here).
In grandparent's case, the issue is not underwater options; they are about to exercise the options and get stock.
At the time they left the company, the stock price was presumably way above the ~$1 strike price. They could have bought (say) $60k worth of stock for only $30k, making a profit on paper. The problem is, as the article says, you have to pay taxes on capital gains AND that stock is illiquid (you can't easily sell it if at all).