Why does overspeed cause structural damage or failure when limit load factors are not exceeded?
Flying way too fast is an example as shown by the graph. For example descending with full power while maintaining 1 G load factor. Or with a powerful enough engine, exceeding the speed limit in level flight.

The air impacts dynamic pressure on the aircraft, the faster the plane, the higher this pressure—up to the design limitations.

What causes structural failure when the aircraft is merely flown fast?

High speed induces vibrations, flutters, etc., all would likely cause loss of control and/or structural damage/failure.


  Flutter: At its mildest this can appear as a "buzz" in the aircraft structure, but at its most violent it can develop uncontrollably with great speed and cause serious damage to or lead to the destruction of the aircraft.


(Source) Flutter.