Unlike non-transgender (cisgender) male and female identities, transgender identities are sometimes contested, and on Stack Exchange sites this would detract from their user experience or may deter them from using the sites entirely. Stack Exchange are taking a firm affirmative stance towards transgender identities in an effort to be more welcoming (or perhaps because they simply perceive it as the "right thing to do").
Every Stack Exchange site uses pronouns (some more than others). However, pronouns for transgender people are sometimes misused (perhaps accidentally, perhaps deliberately), so Stack Exchange seem to be codifying behavior protocols concerning pronoun usage. As many users have stated, pronoun usage ideally would be a matter of common sense and basic courtesy (and not require codification), but society is not at this stage yet---some people either don't know or choose not to use the correct pronouns, often due to inexperience interacting with transgender people (there's not many around).
Analogously, a moderator might step in if a woman user was repeatedly called "he": we wouldn't be particularly concerned if it was a one-off blip, but repeatedly and deliberately calling a woman user "he" could reasonably be considered bullying and grounds for disciplinary action. My impression is that the Stack Exchange team are aiming for a equivalent stance towards transgender pronoun usage, but recognizing that it needs codifying because of a general lack of familiarity.
There seems to be some blowback, and the loudest counter-argument appears to be in the form of compelled speech because the new CoC describes explicitly avoiding using someone’s pronouns as a violation. And the answer to Q9: Do I have to use pronouns I’m unfamiliar or uncomfortable with...? seems to be a misinterpretable way of saying "don't misgender people". However, judging from the new CoC (You are not required to insert pronouns where you otherwise would not) it seems perfectly fine to simply not use pronouns at all if you don't want to (although my reading is that one should avoid selectively choosing users for non-gendered language). All in all, it seems it will take some back-and-forth before the dust settles.
Other issues are about the mechanics (e.g. editing posts to remove stipulated pronouns; how to determine which pronouns to use), how to identify and react to a hypothetical bad actor, issues with (less familiar) singular pronouns such as "they" (particularly for non-native English speakers), and contradictions with non-transgender people's religious belief (although transgender people also have religious beliefs).
My impression is that this whole thing is overblown, with people wildly throwing around words like "Orwellian". I recommend giving Stack Exchange some time to sort out the implementation, time for them to give well-thought-out responses to concerns, and stop intensely badgering them with endless "you suck; I quit" posts. Patience.
Ordinarily, we don't initially know users' pronouns, and opt for gender-neutral language like "OP" or "@username" or "the user". I find it hard to envisage a scenario where a user could misgender someone without engaging in "pronoun guessing" (or doing it deliberately), which is not appropriate even for non-transgender users. A trans person (like me) who prefers "she" will probably make it obvious (like me!) rather than repeatedly say "I use she/her pronouns"---most people get my pronouns right, and don't even realize I'm trans. Relax---you're probably "pronouning" correctly already.