The enzyme is RNA polymerase that replicates the viral genome (RNA viruses, like influenza) and it's responsible for antigenic drift because it has no proof-reading mechanism. It makes a mistake every 1-8 x10^-3 times it replicates a nucleotide. That's not super often, but often enough to cause a problem.
If those mistakes happen in the genes that code for the surface proteins of the virus, then at some point the immune system that would no longer recognize the virus (no immune response!).