That's an illogical statement on the face of it.
Not saying its not true, but evidence or reasoning for why the rest of the population will be protected has not been presented (in this sentence or in those which follow in the article--unless I missed it).
Eg: If a disease has an R0 of 2.0 (ie infects 2 people for each person that gets it), but 70% of the population is immune either because of vaccine or already has had the virus and has antibodies, the R0 is then reduced by 70% (2.0 * (1-.7) = 0.6). Since the R0 is then lower than 1, the virus dies off as the replacement rate is below the stock rate. Simple math.
It has to do with the chain of transmission. If an infected person spreads the virus to two other people when there is no herd immunity then that rate is lowered as the herd gains immunity. What may have been transmission to two people now becomes transmission to one or less.
Maybe a more mathematically inclined person can explain it better.