Can someone explain to me what's the big deal with pronouns and/or gendered words?
OK, my opinion.
A person who transitions socially (= asks for different gendered words, at a minimum) is trying to decouple their social role from the one handed to them by the way our culture processes biology. Often, the social transition is accompanied by other modifications, such as in: body language, clothing, friend groups, hormonal makeup, surgical intervention, etc.
My first point is that on the part of the transitioner, this all represents a lot of effort.
Now, why is it so difficult to be trans? You can look up the statistics on Wikipedia, but what we know is: trans people who are affirmed in their social transition tend to be as healthy as anyone else; trans people who are not so affirmed experience a lot of negative consequences (the 40% suicide attempt rate is one key statistic).
OK, let's try again:
how is it that people push back against attempted social transition?
One way is to punish trans people, often brutally, for gender non-conforming behavior, by which I'm generally referring to street violence, rape, bullying, other forms of direct violent attack. I'm going to call this "second-person violence," meaning that "I" am attacking "you."
The thing is that if it were just random violence, we probably wouldn't have too much of a problem. You kill a man, you are a murderer, you go to jail (ya know, in theory). So how do we justify this direct form of violence socially?
Another way to push back against trans people is to claim that transition is impossible and deliberately deny the trans person's chosen identity. This is "third-person violence," so that "I" am attacking "them." More overt forms of attack along these lines include taking policies that deliberately exclude trans people from social spaces, for example, denying trans women access to bathrooms because "they are actually men."
At the most basic level, the biological fundamentalism tactic involves repeatedly denying an individual's gender in public. It becomes apparent due to the inconsistent social narrative surrounding the person -- for example, we know that Caitlyn Jenner is often called both "she" and "he," but this is not the case for, say, Mike Pence -- that that person is trans or perceived to be a trans ally. For the latter point, I'm thinking about the period when people asserted that Lady Gaga was a trans woman.
The reason why we consider violence against trans people to be structural is that our culture is set up so that people can participate in things like this misgendering by accident. It is the norm.
Repeatedly calling someone of Anglo-Saxon descent by an incorrect name reflects badly on the speaker, who is clearly bullying the other person. But repeatedly calling a trans person by an incorrect name is considered to be an honest mistake "that anyone could make."
Through this process, trans people become hypervisible. Our social identities are denied, hence our ability to participate normally in the community, and we become clear targets for those who prefer direct forms of violence.
This social denial has a cascading effect as our identities are constructed as illegitimate, therefore resulting in: 1. denial of ordinary forms of protection in the face of violence; 2. creation of "reasonable" social policy to continue marginalizing trans people. These are the conditions which are required to make first-person violence 1. "random" 2. "reasonable" 3. "legitimate."
The reason why you are being asked not to assume the gender of strangers, therefore, and to use a person's chosen pronouns, is because even if you personally don't wish to participate in the violence against trans people, those who do want to indulge in this cultural ritual need you to signal your willingness to look the other way.