Its pretty simple, really.
1) This asshole nammed Wakefield ran a criminally negligent study that made up a lot of shit to help a client sue a vaccine manufacturer. Is slipped into a peer reviewed journal, it got some celebrity support because the idea of something so common being so dangerous and explaining something so poorly understood (Autism) appeals to people, and once that happened, any proof to the contrary sounds like a coverup, because people are really bad at scientific methodology.
2) People were willing to throw vaccines to the wolves because the threat doesn't seem real. If you grew up in the modern day and age, Polio doesn't seem like much of a threat because of vaccines. People have a hard time understanding the scope of the horror of preventable illnesses because they have been largely forgotten in the developed world.
3) In a sense, there is a logic here. Vaccine do carry a (Negligible) risk. You can get terribly sick, or die, from vaccines. The odds are practically zero, they are NOTHING compared to pre-vaccine times, and you can't get Autism, but its these kinds of dangers that prime the imagination. If your child doesn't get a vaccine...then chances are, nothing will happen. You have eliminated a negligible risk, and still have the advantage of herd immunity. If everyone else is immune, then you don't really have any vectors for infection. That makes anti-vaccination a tempting philosophy, because if you alone follow it, theres really no risk. But once more people ignore vaccines, the odds of infection increase exponentially. At first, there appears to be no downside, and people can project whatever terrible ideas they have regarding vaccinations without fear, but eventually momentum builds and people start dying. First its the non-vaccinated kids, and then the vaccinated kids (In MUCH lower percentages) because vaccines arn't 100%. But at this point, you have invested yourself in the worldview
Anti-vaccination is sort of like a pyramid scheme: It pays off initially to the few (Though the payoff is practically 0), but as it builds, it gets exponentially worse for everyone. Eventually, people are trying to eliminate a 1 in a hundred million chance of an allergic reaction at the cost of a 1 in a hundred chance of dying from mumps, and passing that infection on to your neighbor. Since people are very bad at distinguishing between a small but significant chance and an infinitesimally tiny chance, their cost/benefit analysis for vaccines gets thrown horribly out of whack, and theres a lot of room for error there to amount to a lot of dead kids. Thats the reason why it works that way. Its still absolutely unacceptable.