This isn't really true. Let's say a company has 100 shares and they decide to issue a $1 dividend. If there is no shorting, the company just pays out $100 and everything is done.
But let's say there is shorting. Someone loans out 10 shares to a short who then sells them to someone else. All of a sudden there are 110 shares out there. So when the company pays out the $100 it's $10 short of what is required for each stockholder to get the dividend.
So where does the required $10 come from? From the party shorting the stock of course! Anyone short a stock is required to pay out, in cash, any dividends issues while they hold a short position. So it all works out.