A "convertible note" is a cash investment that "converts" into an equity investment when the company starts doing equity investments. The have two separate mechanisms for rewarding investors for getting in early.

The "20% discount" means the investor can convert the note into shares paying 20% less per share than later investors.

The "$5 million cap" means, if the company has a valuation greater than $5 million, the investor can convert the note into a fraction of the company as if the company was valued at $5 million.

The investor then chooses whichever of these options is better.

Take the example of a $1 million as a note with a $5 million cap and a 20% discount, when series A funding comes along.

If Series A values the company at $4 million, the investor can choose between taking shares at a 20% discount (they get shares valued at $1.25 million for their investment of $1 million) and taking shares at a $5 million valuation (they get shares valued at $0.8 million for their investment of $1 million) and they obviously choose the former.

If Series A values the company at $10 million, the 20% discount still lets them get shares valued at $1.25 million for their investment of $1 million - but now taking the shares at a $5 million valuation lets them get shares valued at $2 million for the same investment. Obviously they choose the latter!

A $6.25 million valuation is the crossover point, where the 20% discount and the $5 million cap give the investor the same number of shares. At a valuation below that the discount is the better choice, and at a valuation above it the cap is the better choice.