I searched your italicized quote there. First result.
It looks like he's talking about Reye's Syndrome, a pathology that can cause substantial brain damage (and/or other things: Liver damage, death, ...) in children - adults generally recover fully after a couple weeks. (I wanted to be sure he hadn't signed on to the immunization/autism claims, which have been thoroughly discredited.)
Reye/Reye's is a reasonably rare side effect of several viral illnesses, including immunizations for them. Risk of it seems to be multiplied by a factor of something like five if aspirin is taken, but aspirin (or other salicylates) is not necessary for its occurrence. It seems also to be associated with pre-existing metabolic disorders, so some families might be at very high risk while others effectively immune.
It's clear from even the soundbite posted: Rand's claim is that the decision to risk a child's health is properly the parents', and the government should not be able to force the child's exposure to a series of these risks over the parents' objections - informed or otherwise.
Immunizations are partly about population immunity - reducing the density of people susceptible to a disease to the point that it peters out in a declining exponential rather than blowing up in an expanding exponential, thus also protecting those not (yet) immunized, for whom the immunization was ineffective, or who were at risk despite the availability of immunization (e.g. AIDS sufferers). So risk/benefit calculations are for populations as well. Accepting the risk of the immunization helps others as well as the immunized person, so being immunized is partly an altruistic act.
Rand's point is that he believes the government shouldn't have the power to FORCE people to risk their lives for the benefit of others, that these life-critical decisions are personal and should be left up to the people in question (or their guardians if they're too young to make the choice themselves).