There has been quite a bit of research regarding the people who believe in conspiracy theories. It's some pretty interesting stuff, and I suspect much of it translates to the anti-vax crowd. They tend to think they're smarter than they really are, they have few accomplishments, and a desire to feel special. So, they have the 'special knowledge' or ability to 'read between the lines'.I will never understand these Anti-Vaxxers.
Basically, their brains are broken. They'll do things like point to a real conspiracy that did happen and use that as justification to believe whatever conspiracy theories they decide to believe in. They rely heavily on logical fallacies - such as, "you can't prove it isn't true". That's how you end up with an armed moron storming a pizza shop to free the imaginary captive children.
Another grand one is the predictions they make. None of it ever happens, but they just keep making new predictions - and will go so far as to deny their previous predictions. If you quote their previous predictions they say things like you're taking it out of context - even if you quote them directly. Sometimes they'll deny they even made the prediction - even if you show them a screenshot of them making the prediction. "You fabricated that." Which is shortly followed by, "You can't prove you didn't fabricate it!"
I'd keep going, but it'd start bordering on politics and I'm going to avoid that. We'll just say that I could be reasoned into supporting some pretty drastic measures.