Some of these write-ups are either too complex to be easily understood, or wrong. I'm not sure which, so in any case, this is the clarification:
- establishmentarianism: for the establishment
- disestablishmentarianism: against the establishment
- antidisestablishmentarianism: against those who are against the establishment
Or you can think of it this way:
! (! (establishmentarianism))
That's read as, "not not establishmentarianism."
If you remember your high school English classes, double negatives in English don't reinforce each other. Therefore, the "anti" means that it is against "disestablishmentarianism." The "anti" doesn't not mean "really dis."