Why are feminine pronouns used for GMs in rule-books?

Female pronouns are used to be gender-inclusive.

First off, singular "they" as a gender-less pronoun is não ainda combinado sobre as standard English again (see the next paragraph for why I used the word "again" here). It's cropping up more and more in spoken English, but is less common in writing, especially published writing (as opposed to communication such as email).

The use of "they" as a gender-neutral singular pronoun has a long history in the English language. It shows up in Shakespeare and Chaucer, for instance. In 1850, however, the English Parliament declared "he" to be the correct gender-neutral singular pronoun, and was largely successful at getting grammarians to back them.

Thus, for 150+ years masculine pronouns have generally been the default for persons of unknown gender in English. In the past few decades, feminists have begun to push back against this, with attempts at finding pronouns that don't assume unknown persons are male accelerating over time (which is part of why singular "they" is becoming a thing; the rising public awareness of persons who do not identify with either gender is also a factor). If you look at slightly older RPGs (e.g. D&D 4e, 13th Age), many use "he or she" everywhere to avoid specifying a gender.

The RPG hobby in particular was heavily male-dominated when it began. As RPG publishers have come to see the value in attracting female participants (both as players and GMs), they have tried to make their books more inclusive in an attempt to expand their customer base. Another example of this trend is the shift away from art depicting scantily-clad women being rescued by all-male parties, which was not uncommon in RPG manuals of the 1980s and early 1990s.

Using female pronouns for readers of RPG manuals is an easy way to express that it's perfectly normal/OK for women to play/GM RPGs. I have never seen any evidence of this being a negative for male readers of these manuals, who generally don't have trouble envisioning themselves participating in RPGs (since, again, the hobby has been male-dominated for most of its existence). Paizo uses female pronouns for its female iconic characters; per their creative director a desire to address inequality is a significant factor in that. White Wolf and its offshoot Onyx Path have been particularly up front about their desire to make their games inclusive. If you go looking for examples, you can find a lot of women talking about how this stuff really faz a diferença para eles.

Some rulebooks even explain this directly in the text. From the All Flesh Must Be Eaten core rulebook, p19:

Every roleplaying game struggles with the decision about third person pronouns and possessives. While the male reference (he, him, his) is customarily used for both male and female, there is no question that is not entirely inclusive. On the other hand, the "he or she" structure is clumsy and unattractive. In an effort to "split the difference," this book uses male designations for even chapters, and female designations for odd chapters.

Of course, some publishers disagree with this change and still use masculine pronouns, and, as above, a few will directly tell you why they're doing so. From the CthulhuTech core rulebook, p13:

Okay, here it is – we use he, him, and his when we’re talking about people playing the game. It just seems weird to alternate pronoun genders within the same book – it makes it feel like the book is written for two different audiences. The masculine pronoun is the standard and right or wrong we’re used to seeing it. It may not be politically correct, but you can’t please everybody.

As for why English-language RPG books are doing this but French-language RPG books are not, that's harder to say. One possible factor is differences in how "correct" usage of the language is defined. The English language has historically been heavily on the "descriptive" side of defining language (especially in the US, and especially in the past 50 years or so); if enough people use the language in some way, then that's correct English. The French language, on the other hand, has tended towards being defined "prescriptively"; the Académie française is oficialmente e legalmente in charge of defining what is, and is not, correct French language usage. This may make French language publications slower to change existing trends of pronoun usage. Additionally, French is a gendered language (i.e. all nouns have a gender), while English is not. Another possibility is that the feminist-led drive to question the "default masculine" in the US has not taken hold to the same extent in French-speaking nations.

In addition to the current tendency to be gender inclusive, using different pronouns helps to clarify different roles when a manual, role playing or otherwise, describes interactions.

We see it in different contexts for example if a software manual or use case has to distinguish between a user and a software developer: is will use the male pronoun of for one role and the female pronoun for the other.

If we describe a rule in role playing about an interaction between the DM and a player with a single pronoun it can become quite confusing:

He declares the action his character takes, then he rolls a die on table 4.27 to determine the outcome of the action the player declared.

Instead, if we have previously established that we use the male pronoun for players and the female pronoun for the DM, we can say:

He declares the action his character takes, then she rolls a die on table 4.27 to determine the outcome of his character's action.

With different pronouns the interaction is more clear than by using he, she, he/she, or they for both roles in the interaction. Of course, always using 'the player' and 'the GM' is also possible but it reads somewhat more difficult.

Paizo has declared their company to be gender-inclusive

While the other answers have already discussed how and why the game industry have been using less and less male pronouns in their books, we know that Paizo developers have tried to be inclusive in their texts for over a decade now.

We have over half the iconic characters to be females, we have dark-skinned iconics, native-american-inspired iconics, homossexual iconics, bissexual iconics, asian-looking iconics, obeso iconics, underaged iconics, idade iconics, and even a transgender iconic. There are lots of people that dislike this inclusiveness in their games, and this has been disclaimed in some of their books (namely, Starfinder and Pathfinder 2nd edition) and caused some irritation in a small part of the community (which, in my opinion, is a good thing).

Não faz muito tempo, James Jacobs (Creative Director at Paizo) has answered a question about asking why there were female pronouns in their books, to which he had to explain that the female pronouns are used when the iconic character is of female gender:

Why are all of the books written in female-specific pronouns (she, her) and not just simply gender-neutral ones?

The books aren't written in female-specific pronouns... I suspect you're only noticing those because you're used to seeing male pronouns.

When we write about a specific class, be it the class description or an archetype or whatever, we use the pronoun as set by that class's iconic. SO... when talking about fighters, we use male pronouns (because Valeros is male). When talking about sorcerers, we use female pronouns (because Seoni is female).

Anyway... the fact that it's unusual to use female pronouns is all the reason I need to keep using them, because that means there's some inequality going on. When the switch between male and female pronouns ceases to be surprising, that's where I feel that society needs to be.

Mas they have clarified that they follow the Chicago Manual of Style for pronouns, which suggests them to this approach they adopted. Instead of using eles for a known person, use he or ela whenever the author feels appropriate, respecting the person's preference. So, if a male wants to be addressed as sua, that is what the author has to go with.

This does not explain, however, porque they decided to use sua whenever the gender is unknown, such as on nearly all references to jogador or GM in their books. But we do know that they decided to do so because gender neutrality in English is confusing, wrote it down in a document for future reference, and have been doing it ever since.

Mas desde Inglês is, indeed, difficult to handle gender-neutrality, authors will have to pick one method, generic He, Ou o generic She that sounds non-sexist but will actually bring up why He wasn't used, or the singular They that will cause more confusion than it tries to fix. Whatever is chosen, there will be someone to complain about it, and it explains why we keep seeing rulebooks using different pronouns like you noticed.

In the past, technical literature in English used "he" to refer to a person of any gender. This masculine assumption is now considered sexist and non-inclusive.

Writing "he or she" is possible, but general opinion is that this is harder to read. Common practise instead is that where an example of a user has to be given, the author alternates strictly between male names/pronouns and female names/pronouns. This ensures the text is inclusive.

French has the option of using "on" to represent "some person". However it does not have ungendered nouns, so "the GM" must be either masculine or feminine. Examples should therefore follow the same rules of inclusive writing as for English, where they are written by companies with an inclusive-language policy.