There is a visceral reason for people not wanting to run the risk of vaccinating their children, even though the risk of damage caused by getting a disease instead is greater.
If I give my child a vaccine and he or she then has a reaction which causes damage, that is my fault: it happened because of something that I did.
If I don't give my child a vaccine and he or she then has a disease which causes damage, that is not my fault: it didn't happen because of something that I did.
That is completely nonsensical when you actually stop and look at it, but it is the gut feeling some people are certainly going to have. Better to do nothing and run a risk, than do something and run a far lesser risk; it's a very standard human reaction, that one, better the devil you know. It is why people go on living on the slopes of volcanoes: it may not happen, and if it does, well, it isn't their fault.
Oh, and "anti-vaxxers aren't stupid, my anti-vaxxer friend is a rocket scientist/Nobel prize winner/best-selling author" is making a bit of an assumption: it is entirely possible for a respected professional to be extremely stupid about things outside his or her immediate field of expertise. Many very bright people are. The phrase is "all that brain and she doesn't have the sense to come in out of the rain." Is your friend's PhD in immunology/epidemiology? if not, their opinion on those is likely to be no better than that of any layman. (And if they are Andrew Wakefield, they have been well-paid to have a particular view and spread it.)