Vaccines work by making us produce antibodies to fight disease without actually infecting us with the disease.
If a vaccinated person then comes into contact with the disease they were vaccinated against their immune system will recognise it and immediately produce the antibodies they need to fight it.
But if they come into contact with another disease/pathogen their immune system's memory cells don't recognize it therefore they don't have any tactics to fight the pathogen. Therefore you get symptoms such as the chicken pox spots.
p.s: there isn't a chickenpox vaccine yet
