I think it comes down to how anti-vaxxers receive and interpret information, as well as approaching things from the perspective of "I am smart and can see what is real better than most." This applies to medical dignoses of any kind, politics, culture. And it gets amped up when you have kids: what toys are best for my kid, what's the best diaper, which sunblock is safest, etc.
I live in SF and have a son and see this mindset everywhere. Everybody with the Honest diapers. Everybody with the Uppababy stroller. Everyone is doing all their own research online and asking friends who do the same, and it becomes a tight feedback loop where there is one right answer to everything about how to raise their kids. And the tricky thing is, everyone feels like they did the "research" themselves, so they can trust it.
The anti-vax movement is fundamentally anti-intellectual and promoted by people who should know better (and who pride themselves on knowing better). And to answer your question, people get angry because each anti-vaxxer increases their own children's risk for dying needlessly from a terrible disease.
There is no question that the antivax movement was started by an idiot fraud report and was pushed forward by Jennie McCarthy (who did her own research). She has since reversed her stance, but to support their established belief people keep drumming up new reasons vaccinations are bad, but it all boils down to a bunch of ZOMG TOXINS fearmongering.
I boil it down like this: your child is thousands of times more likely to die from catching the measles in Marin than having a terrible reaction from getting a vaccine. And vaccines are a tiny risk that is spread out amongst the community to protect us all. It is a civic duty to get vaccinated. It is the tiny price of living in a civilzation and to reject it is to say "A .0001 risk to MY child is not worth protecting everyone else's child." It is fundamentally selfish and fundamentally anti-intellectual which is why people are mad at you.