@Jeff L. Looks like I have a different version of the book than you (mine is copyright 1994, 2013). The scenario you described appears in a slightly different format on page 62 of my copy. It begins with this premise in mind "The true rate of interest you are paying depends on the amount of discount the person you owe would give you."
Taking the example of "Johnny" (in my book it just uses "you" and "me") owing you $10,000 at a stated interest rate of 10% with payments of $132.15 and 120 payments. Although @Gail Greenberg brings up a good point that what you originally paid for the note will make a big difference, in this case, the hypothetical example is written assuming you originated the note to Johnny and he actually owes you $10,000. The point the book makes is that if you would not give Johnny any discount if he paid you off today, then the interest rate would, indeed, be 10%.
Then it goes on to look at the case where you tell Johnny you will give him a discount and his debt can be satisfied if he pays you $8,000, provided he pays you in three days. What is the interest rate that Johnny is actually paying if he takes you up on the $8,000 payoff? N=120, PMT=-$132.15, and now PV=$8000. If you solve for I, you get the 15.6% you mentioned in your original post. The whole point of the example was that you should never trust the numbers written on the face of the documents because as soon as you change one, the others change. The numbers are a matter of negotiation, and you should be negotiating to see if you can change them in your favor.
As to why you might offer this type of discount, it hinges on the fact that you have placed a short time limit on your offer. You don't know if Johnny is about to sell the property, in which case you would be getting a full payoff out of the closing proceeds anyway. Napier argues that if someone asks you for a discount if they pay you in full, giving them a short timeline may end up starting a dialogue with the borrower about what is happening, and it reduces the chance they will actually pay you off short. He says most people would miss the deadline for the discount rather than raising the money.
If Johnny does end up paying you off early, then it goes back to Gail's point about velocity of money. You have $8000 NOW to re-invest in something else.