Rights Issue is done by the companies in order to fresh capital, not by going public though. It rather offers the shares to only a selected group of people, i.e., their existing shareholders at a discounted price. They are offered in a ratio proportionate to your shares. For example- Rights issues are offered 1:5, which means for every 5 shares you hold, you get 1 share.
How does it affect the stock price? A company issuing rights shares gives an indication of new development in the company. There could be various reasons for a company to issue the right shares. More the shares are offered to the market, the share prices are likely to get diluted and tend to go down.
However, one should not always invest just because a share is offered at a discount. Always take consideration as to why the company is planning to launch Rights Issue in the first place.
Hope this addresses your query!