This website seems to mainly be talking about naked short selling, which is selling a stock without first owning or borrowing it. This not necessarily illegal, although can be under some conditions, including aggressive short selling. Wikipedia has a fair amount of info on the practice, https://en.wikipedia.org/wiki/Naked_short_selling
The practice of a selling a share that one does not own is supposed to be used by market neutral market makers to ensure their is always liquidity. They must then at some point in the future buy a share from the market to replace the phantom (i.e., counterfeit) share that they sold.
This practice is useful when there are suddenly more buyers than sellers and the market maker exhausts their standard supply of shares. Otherwise there may not be any bids on the stock at any price for sometime. Other argue that this is no longer necessary with many participants beyond market makers that also employ algo trading and can quickly adjust to increasing demand. Further, if lack of supply becomes a regular occurrence, investors may start regularly setting high, possibly even ridiculously high, standing sell limit orders such that there will always be bids at some price points.