@freeyang,
I'm not a lawyer, but in plain English it means that whatever right is being discussed lasts until the end of the period of the copyright. When the copyright ends, so does the right.
For example, the author has the right to be paid every time somebody publishes his book. That's the point of having a copyright. That right to be paid runs out as soon the copyright on his book runs out-copyrights don't necessarily last forever.