TLDR:
A conversion discount gives convertible note or SAFE holders the right to convert their investment into equity at a lower price per share than what new investors pay during a subsequent financing round.
How Conversion Discounts Work in Practice
If an investor puts $100,000 into a SAFE with a 20% conversion discount and the company later raises a Series A at $1.00 per share, the SAFE investor would convert at $0.80 per share (20% discount), receiving 125,000 shares instead of 100,000 shares — 25,000 more shares than a Series A investor investing the same amount.