Maybe folks are skeptical about the fraudulent NHTSA complaints? If so, here's a source: https://electrek.co/2016/06/13/tesla-fale-complaints-suspens...
The answer is that a short sale is borrowing a share and then selling it, which increases the number of shares available for sale and by supply and demand reduces the price (until the short seller eventually buys back the stock to return it to the person they borrowed it from). In the meantime, which can be as long as the short seller is willing to pay interest, the stock trades lower because the extra share is on the market. It's essentially the same principle as the way banks lending money to people causes inflation because it increases the money supply and so lowers the value of the dollar.
Seeing a lot of existing shorts also tends to spook buyers, which also lowers the price, though it also attracts buyers hoping for a "short squeeze" that raises the price when the short sellers eventually have to buy back the stock.
I saw that other answers had already explained how shorts lower the stock price, but nobody had pointed out the other ways in which they "damage the company", as the original post put it.