Each shareholder receives one right per share held on the record-date. A stated number of rights plus cash buys one new share at the subscription price. If you exercise in full your percentage ownership is unchanged; if you do nothing you are diluted, though in many markets the rights themselves trade and can be sold to recover the value.
The share price mechanically adjusts to a theoretical ex-rights price on the ex date, in the same way it adjusts for a dividend. A headline "50% discount" is therefore much less generous than it sounds.
Example: a 1-for-4 rights issue at $6 on a $10 stock. A holder of 400 shares may buy 100 more for $600. The theoretical ex-rights price is (400 x $10 + $600) / 500 = $9.20, so each right is worth about $0.80.
Related: dilution, record-date, offering-discount, follow-on-offering, ex-dividend-price-adjustment