When you make a clear point again, you re-emphasize it. Politicians running for office tend to re-emphasize the same points over and over....and over again.
When someone re-emphasizes a point or idea, they've expressed it at least once before, and now they're accentuating its importance. You could complain about potholes by calling the city hotline, and then re-emphasize your unhappiness with the roads by writing a letter to the editor of your local paper. Re-emphasize adds the "again" prefix re- to emphasize, from the Greek emphasis, which was used in rhetoric to mean "significance" or "indirect meaning."