Reverse Greenshoe Option
  
An IPO is kind of like the moment when a parent takes the training wheels off their kid's bike for the first time. They run alongside the bike for the first few feet, but eventually let go, standing proudly in the street as their kid rides off into the distance.
The reverse greenshoe option is there in case the kid runs directly into a parked car.
In an initial public offering, a company lists its stock on a public exchange for the first time. Typically, there's an offering price, which is the price at which the stock first goes public. Then the shares are off, trading moment to moment, day to day, like any other stock. (That's the "let go of the bike" moment.)
A well-managed IPO will see shares rise early in their trading career. Market demand will be higher than the amount of stock offered. No one wants their stock to go public at $15 and instantly drop to $13. It shows a lack of confidence by the market, and doesn't allow much room to pay off those folks who got in on the deal early, buying shares at the offering price.
Ideally, the stock will see an initial bump...go public at $15 and rise to $17. That situation evidences a much healthier supply/demand dynamic.
A reverse greenshoe option is used to support the share price in case demand for the stock turns out to be lighter than expected. The provision allows the underwriters of the offering (the people running the IPO) to sell shares back to the company. Basically, they can take shares off the market and send them back to the company. The lower supply makes the existing float more valuable (because there are fewer shares to go around), supporting the stock price.