This is a simple enough phrase, but not one that everyone who is anti-vaccine could have arrived it independently. It must be a phrase that is used frequently on alternative or anti-vax websites, then adopted by all.
It's an odd phrase, because in one sense we all are self-taught, acquiring knowledge only when we are putting in some effort. It is not only geniuses who are autodidacts. Yet there is a real sense in which none of us has done much to "educate ourselves." We stand on the shoulders of giants, and could not reconstruct even 1% of current knowledge starting from scratch.
This comes up because my youngest cousin asked on facebook whether she should have her 12 y/o daughter get the HPV vaccination and got an earful. She quickly deleted the entire thread (for which I congratulated her.) Several commenters used variations of "you have to educate yourself."
It must have powerful juju to be so meaningful to so many who have heretofore researched nothing, but now devote hours finding examples of what I they want to hear. If this seems an unfair generalisation, it might be. I can see rejecting any number of individual pieces of modern, Western medicine. But I have no one in my experience who researched other things before coming up against the vaccination question and then rejected them. My anti-vaxxers, at least, are all people who may be smart enough in some ways, but never showed any interest in looking things up before.
"Educate yourself," then, appears to mean "listen to only one side of things." Powerful.