The measles vaccine is live attenuated, so in the rare case it does cause disease, it's less severe than wild type.
Measles is one of the most infectious diseases known to man. In a fully susceptible population, one infected person can expect to infect 14 other people. The reason only "the few hundred people" get infected annually now is because of widespread vaccination campaigns.
Measles used to be all but inevitable, infecting literally millions, and killing hundreds each year in the US. The only reason we have come to fear vaccines is because we as a society no longer remember fearing these diseases.