Most people have a natural inclination towards fairness and inclusion, even if it manifests differently in the political approaches we take. It's easy for a neutral party to take the TRA position, because the overall message is simple and positive. Let these poor suffering people be treated well.
And if you take that viewpoint, it's difficult to understand why anyone wouldn't go along with this. Do those TERF-y feminists want to make this minority's lives difficult? Can't they just compromise, just a little? Surely there's a simple way to keep everyone happy, if only those opposed cooperated. But the TERF-y feminists continue to oppose, so they're obviously doing this because they lack compassion.
But if you try and dig deeper, the TRA position falls apart. And on some level the TRAs know this. They believe trans people's well being and even their very lives depend on them winning the argument, but they can't win it. And so they panic under this pressure, and their panic becomes emotional outbursts and personal attacks. They don't quite understand why they seem to be losing the argument, because it's always a good thing to want people not to suffer. Surely anyone who opposes alleviating suffering is a bad person? Maybe if the "anti trans" people have a point, it means those who seemingly lack empathy for e.g. refugees, the homeless, benefits claimants, have legitimate counter-arguments that aren't predicated on being horrible human beings. Or maybe humanity is really that cold and evil. Neither realisation is particularly appealing for hopeful progressive types who dream of a unified Star Trek future.
For more middle class people like Zoe Williams, I think there's an additional resentment of privilege. If you're white or straight, you have majority privilege. There are others on your team who are like you and have your back, and you have a certain social influence over the minority who aren't like you because they need the rest of you for allies. But if you're posh, you have minority privilege, and you can't reconcile your politics with your background. Worse still, your parents couldn't help being straight or white. But they could help sending you to private school. Note how resentful the linked article sounds; she would never make such a terrible choice, it doesn't really matter that she was privately educated because she had such an awful time. It's implicitly her parents' fault.
Furthermore, I think some left-wing people from very privileged backgrounds also have an unconscious saviour complex. Many have always been divided from the working and lower-middle classes in some way. They haven't had to work long hours in smelly factories or scrub poo off a floor for minimum wage. They might see what life is like for the common oiks, but they don't fully get it. Nor do they truly understand why those common oiks might reject policies and ideologies that, from the privileged outside perspective, would improve their lives and make them better people. Maybe they're just uneducated. And so education becomes the focus, the mission, the passion.