Learn English – Why do American and British English use different quotation marks

This is simply a question of style. Wikipedia has a huge article on the subject. The passages most relevant to your question are:

Quotations and speech
Single or double quotation marks denote either speech or a quotation. Neither style—single or double—is an absolute rule, though double quotation marks are preferred in the United States, and both single and double quotation marks are used in the United Kingdom. A publisher’s or author’s style may take precedence over national general preferences. The important rule is that the style of opening and closing quotation marks must be matched[.]

[...]

Use–mention distinction
Either quotation marks or italic type can emphasize that an instance of a word refers to the word itself rather than its associated concept. [...] A three-way distinction is occasionally made between normal use of a word (no quotation marks), referring to the concept behind the word (single quotation marks), and the word itself (double quotation marks). [...] In common usage, there may be a distinction between the single and double quotation marks in this context; often, single quotation marks are used to embrace single characters, while double quotation marks enclose whole words or phrases[.]

Emphasis mine. Read the entire article for further insight.

The phenomenon you describe is almost certainly an artifact of the default settings for apostrophes/single quotation marks that Tor Books' word-processing program enforces.

Here is an example of how Microsoft Word's default settings regularize single quotation marks within double quotation marks:

“He told me ‘I’ve been waiting for you, Mr. Moon,’ and then he disappeared.”

But here is what you get when the opening single quotation mark appears immediately after the opening double quotation mark:

“’I’ve been waiting for you, Mr. Moon,’ he told me—and then he disappeared.”

The difference between the first example (where the opening single quotation mark looks the way it should) and the second example (where it instead looks like a single closing quotation mark—or rather, like an apostrophe) is that there is a letter space before the opening single quotation mark in the first instance, but there isn't one in the second instance.

The default settings in Word don't distinguish between the apostrophe keystroke and the opening single quotation mark keystroke—which makes perfect sense, given that the same keystroke is used for both functions on most keyboards. Word doesn't have an elaborate assessment program for determining whether, in a particular instance in which the typist types the ' character on the keyboard, the typist intends the ' to be an opening single quotation mark, a closing single quotation mark, or an apostrophe.

To handle the majority of cases correctly, Word by default assumes that, if there is a letter space or line break immediately before the ' keystroke, the typist intends to create an opening single quotation mark; and conversely it assumes that, if there is another character/keystroke immediately immediately before the ' keystroke, the typist intends to create an apostrophe/closing single quotation mark (which look the same in most fonts).

Unless the publisher alters the coding for the default specification or manually changes every instance of “’I’ve [or whatever] ... to “‘I’ve ..., the result will be a book full of backward opening single quotation marks. I can tell you from experience that it's no treat to run a search-and-replace operation for such characters, because you have to wade through a lot of correctly rendered apostrophes to reach all of the erroneous single quotation marks. But there is no easy way to avoid the problem, other than to abandon smart (curly) quotation marks (and apostrophes) in favor of dumb (straight) ones—which many publishers refuse to do.

Update (1/4/16): In his answer, Benjamin Harman makes the interesting suggestion that you can avoid Word's (and perhaps some other word-processing programs') default apostrophe-as-opening-single-quotation-mark problem by adding a letter space between the opening double quotation mark and the following (intended) opening single quotation mark. This would certainly avoid Word's automatic reversal of the direction of the opening single quotation mark, but it raises a couple of new issues.

The first issue is aesthetic: To a reader accustomed to seeing embedded single quotation marks run after double quotation marks in the form “‘I’ve ..., the form “ ‘I’ve ... may look weird, just as the seventeenth- and eighteenth-century inclusion of a letter space before a semicolon does.

The second issue is practical, and more serious: If you're going to add a letter space when the opening double and single quotation marks are consecutive, as in

“ ‘I’ve been waiting for you, Mr. Moon,’ he told me—and then he disappeared.”

consistency would seem to require you to add one in the reverse situation as well, where the closing single and double quotation marks are consecutive, as in

“Before he disappeared, he told me, ‘I’ve been waiting for you, Mr. Moon.’ ”

Unfortunately, Word applies the same rule for double quotation marks as it does for single quotation marks—namely, that if a letter space precedes either punctuation mark, the default instruction is to treat the mark as an opening (not closing) quotation mark. So what you get by default with a letter space after the closing single quotation mark is this:

“Before he disappeared, he told me, ‘I’ve been waiting for you, Mr. Moon.’ “

To counter this default misfire, you still have to go through the entire manuscript and correct the keystroke by hand; or you have to type the two quotation marks without a letter space and then go back through the text and add a letter space between each such pair.

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With regard to Chris H's comment beneath this answer, economic considerations aside, it makes perfect sense to use an expanded keyboard with the necessary keys for opening and closing quotation marks, and/or to use publishing software that makes less ham-handed decisions about what character you mean to type in a given situation. But many publishers (including the computer magazines where I worked for many years) have long since dispensed with high-end typography systems. They write their content in Word, pour the text into layout templates designed in InCopy (or formerly, Quark), and pass the resulting work through an automated spelling check or perhaps a quick human copyedit/proofread in hopes of catching any resulting problems before going to press.

This system emerged for exactly one reason: It's much cheaper than the old system. But one consequence of cutting out typographers (and in many cases, copy editors and proofreaders) is that it introduces errors such as the one that the OP asks about. Tor Books (the publisher of the book that prompted the poster's question) isn't some fly-by-night outlier in the publishing industry, by the way. According to its Wikipedia page, it is owned by Macmillan, which also owns St. Martin's Press, Henry Holt, and Farrar, Straus & Giroux. What we see with Tor Books is the direction that mainstream publishing has begun to take and—as far as I can see—will continue to take.