You're not missing anything if we're talking about an innoculated child. The assumption is that there could be a number of parents that decide not to have their children vaccinated. A child gets sick, then passes it along to their siblings, who then go to day care where there could be other youngsters not vaccinated and the disease is passed along.

Kids get chickenpox, they might not show signs of the illness while they are "shedding" the virus. Older adults might now develop shingles, which can be very painful for many. Adults not vaccinated as youngsters are also at risk for developing the disease, chickenpox, and it can be more serious. People travel, bringing along with them various viruses and again, spread disease even while they might not know they're infected.
While these childhood diseases aren't as prevalent in the US as they once were, they could re-emerge once again, in full force.