Printing prices on books began in the mid 19th century.

There were two reasons for printing the price on a book. The first was advertising. Dime novels in America and penny dreadfuls in the UK are two examples of books that benefited from printing the price on the front cover.

But you weren't asking about printing the price on the front of books. You asked about prices on the back. Prices on the back of books have their origins in something called a "Fixed Price Book Agreement". This is an agreement between the publisher and booksellers which fixes the price of a book. If a bookseller sells a book for less than the price published on the book the publisher will no longer sell them books.

The effect of these agreements, which are also sometimes laws, is that all booksellers are equal and so cannot compete on price. Amazon for example would charge the same price for books as your mom and pop store down the street. This is nice because it allows mom and pop stores to compete against retail giants. However, there is a downside. These agreements/laws mean higher prices on books. There are is much more to the argument Free or Fixed Prices on Books - Patterns of Book Pricing in Europe covers the debate and arguments very well.

Perhaps the most well known of these agreements is the UK's Net Book Agreement of 1899. In place until 1994 this agreement guaranteed that all books printed in the UK would include the publisher's set price printed on the cover.

In some places these agreements are illegal or unenforceable. Often publishers still include the price printed on the book but it is just a suggested price.

If you are interested in reading about the origins of the UK's Net book Agreement from a publisher's standpoint I found an online excerpt from The Net Book Agreement 1899