Stock options represent the option to purchase a share of stock in the future at some specified price, which is called the "strike price". This price will be set when your options are granted, although you will have to wait until the options actually vest until you can exercise them (exercising an option means buying a share at the strike price). The strike price is often set as the market value on the day of the grant, although here it seems the strike price is set by the board explicitly. If the board sets the strike price below the market value, options have immediate value.
The strike price may or may not be a "very low price" depending on what the stock does in the future. If the stock price remains above the strike price, you retain the option to buy the stock at the lower strike price - this is great, since you can buy at the strike price and sell at the market price for an immediate profit. If the market price goes below the strike price, you are left with the option to buy the stock at higher than market price, which one would never do in practice - if the stock price falls below your strike price, you'd be better off buying a share from the open market rather than exercising the option.
The package example shows a rather lucrative stock option, where the market price is higher than the strike price - in the example, you are able to exercise an option to buy a share that's currently worth $12.75 for the low price of $2.60. This does not necessarily mean that your equity package will be worth a similar amount - if the stock price falls below the strike price, a grant of stock options is worth absolutely nothing. Compensation in the form of stock options gives the benefit of large potential gains, but they are risky in that they may ultimately be worth nothing. Having a board-set strike price that's below the market price at grant reduces the risk somewhat, since the options will be worth something the day they're granted, and will continue to have value even if the stock price doesn't move at all.
After options vest, you'll have some period of time (perhaps 10 years, check your plan specifics) in which you can exercise them to buy stock at the strike price. You can immediately sell that stock to earn the profit over the strike price (the $10.15 per share in the example), or you can hold onto the stock like you would any other share, and sell it at a later date to enjoy a lower, long-term capital gains tax (but then you risk the stock falling below what you paid for it).