What characteristic of waves can cause a bridge to collapse?

When the frequency of a wave matches the Natural frequency of the bridge the oscillation of the bridge can build up in amplitude up to the point of causing its destruction. This phenomenon is called Resonance.
You can think of it as the same as when you give an impulse to a person on a swing; if the frequency of your impulse matches the frequency of oscillation of the person, the amplitude of the swinging will increase!
For a bridge the impulse leading to resonance can be produced by intermittent wind, marching soldiers, etc.