Today's clause is a miserable one to be contemplating when your book is new and shiny - indeed, is still a twinkle in the publisher's eye. It's like taking out a pension plan for your newborn, or even planning its place in the family crypt, or adding it to the family tree with its birth date and an ominous hyphen which the death date will one day follow. This clause is about remainders - the remains of your book after its demise.
The Publishers may sell part or the whole of the residue of any edition at a reduced price or as a remainder at the best prices such remainder stock will fetch, the Author having first been given the option of purchasing some or all of such copies at the remainder price, such option to be exercised within 14 (fourteen) days of notice being given to the Author at their latest known address of the Publisher's intention to remainder the work.
Remainders are copies of the book which the publisher cannot sell. Originally, a book was remaindered when bookshops no longer ordered it to keep in stock. Now, of course, a book may sell through Amazon or the publisher's website in a steady trickle, or even occasional drips, and there is no clear point at which the book is no longer selling.
In the best possible case, your book will never go out of print, and will continue to sell even after your death. More realistically, your book will sell for a while and then it will drop from public view, or be superseded by a more up-to-date book. If you are lucky, the print run will sell out and there won't be piles of unsold stock to worry about. These days, a publisher may move your title to POD (print on demand) if the print run sells out but there is still a very small demand for the book. (Whether you should accept a POD edition as in print is a moot point, and one we will deal with on another day.) In the worst case, sales will drop off while the publisher still has a substantial stock of the books. Then they will try to recoup some of the money tied up in the stock, and being wasted on storage, by remaindering the book. This means they will sell it to anyone who will take it at a knock-down price. This clause says they must first offer the remainders to you, at the same price as someone else is prepared to pay. Whether you should take them is a separate issue, and not one for today. Pride (hurt) will tempt you to buy them, but think about what you might do with them first.
Now, the publisher will give you 14 days in which to recover from the shock of being remaindered and then say how many you want to buy. If you are in the habit of going away for more than 13 days at a time, you might miss your chance. I'd suggest that you ask for the publisher either to give you a bit more time, or to contact you at your last know real address AND last known email address. We would hope, of course, that the publisher knows where you live as they are sending you royalty cheques. If you have an agent, ask them to contact your agent, too. I know it's not nice to do this now, and might feel like negotiating a pre-nuptial agreement, but it's only a few moments of pain and may save you substantial disappointment later.
Once you have the remaindered copies, it's up to you what you do with them. You can sell them yourself if you can be bothered. If you do school visits or readings, this might be a very real possibility.
And finally.... being remaindered is not the worst thing that can happen to your book. I remember hearing the following conversation between two authors a few years ago. No names, I'll call them A and B:
A: Hello, B, how are you?
B: A bit glum. My book is remaindered in Galloway and Porter [a now-defunct bookshop in Cambridge].
A: Oh, I'm sorry to hear that. [pause] I wish mine had been remaindered.
A: It was pulped.
Yes, if the publisher thinks they can't shift your book however low the price, they will have it pulped - literally, turned into paper pulp. They have to pay for this, so they only do it if your book is so unsuccessful they can't even give it away and have to pay someone to take it away. You could ask that the publisher adds a clause to say they will give you the remainders if they are planning to pulp them - but that might suggest you lack confidence in your book, so I'd suggest you leave that request until they offer you the remainders for money.