A Rights Issue is a fund-raising exercise by a company. It gives the existing shareholders the opportunity to buy more shares in a company at a fixed price (usually at a discount to the market) on a stated future date.
The company issues new shares and sells them using the proceeds for a specific purpose. It can often be a move to pay down debt in difficult times or it can simply be to fund an acquisition.
The mechanics of a Rights Issue can appear complicated. But in simple terms the company is raising money and is issuing new shares to raise the money. Understanding the reasons for the fundraise is the important element. That should be the primary reason for supporting the Rights Issue shares or not.