Just throwing this out there as well:
A lot of people are just uninformed.
Like, back in high school, my mom met someone who she suspected was trans, and she muttered about how trans women were a "caricature of women," and "trying too hard to be female," and she seriously thought, based on the limited (mis)information that she got about trans people from the media, that they were just men who always liked feminine things, and so they arbitrarily decided to be female so that they could do those feminine interests. So she basically saw trans women, as a lot of people do, as just gay men pretending to be women.
I was misinformed as a teenager too, as a lot of us were. I didn't know about HRT. I didn't know that it was possible to get a (more or less) completely female body from hormones alone, no surgery required. I didn't know that you didn't have to have childhood dysphoria to be trans. I didn't know that you could be trans despite having very tomboyish interests. And I didn't know that there were trans people across the whole gender-expression spectrum, I thought that every trans woman had to be hyper-feminine.
Youtube was what changed this for me, where I finally got to see trans women tell their own stories instead of biased news programs which constantly sensationalized it. I realized it was common, and it wasn't the big flipping deal everyone made it out to be.
And guess what? My mom, who would have been one of those people who spoke out against trans people and didn't really understand why they'd want to be women, learned, because of me. Because I got to explain to her first-hand, in my own words, what it felt like. And she was willing to listen, and so she learned. She's even told me first-hand, "I never understood trans people until you started telling me about it. Now I do."
So that's the real problem here, is just misinformation... the media constantly sensationalizing the transition process, framing trans narratives as a sideshow attraction ("Hurry, hurry, hurry, step right up! See the amazing woman who was born a man but went through surgery to remove her penis! Whooo! You'll be amazed! You'll be horrified! Step right up!") [and this is why trans men are so invisible in the media, because it's a lot less "sensational" for a woman to want to be a man, since our culture has a big misogyny problem.] That is the issue, just that trans people never really had the opportunity to tell their own stories before. But now that we do have that chance, things are quickly changing.
A lot of people simply never have the chance to hear from an actual trans person who they are, what they're like, and what their gender means to them. And for a lot of people, it's simply a matter of talking to us and knowing us where they realize for the first time that we're not those perverse female-appropriating crossdressing homosexual fa****s that the media makes us out to be, we're just normal people trying to live our lives.
Information is our ally. The real question is just whether people are willing to listen or not. And to most people's credit, they will listen. It might take them a while, and it might take a long mental battle where they start at denial but slowly move toward acceptance if the message is repeated enough times. There are always a few stubborn people, who once they get set in their ways they'll never change their minds. But for a lot of others, it's just a matter of getting out there and having a chance for them to meet us and learn that we're pretty normal people, and maybe giving them some time to come to terms with it because it challenges their cultural assumptions so much.
I'm pretty optimistic about this. I think we're winning the information battle right now thanks to the visibility of activists like Janet Mock and Laverne Cox. And where being trans used to be a social death sentence if anyone found out, now it's quickly becoming acceptable, especially among well-informed people who are willing to listen and learn. (We have a LONG way to go before we've changed the culture at large, though. Uneducated people who are uninterested in learning still as a whole treat us like freaks. But then again they're still treating gay people like freaks too, so good luck with that. When people are unwilling to learn, it takes FOREVER, and a HUGE change in cultural perceptions, pretty much every other person they know being okay with it, before they'll change.)