How can "quantum particles have positive masses, even though the classical waves travel at the speed of light"?

Clay Mathematics Institute writes about the Yang-Mills and mass gap problem on this page http://www.claymath.org/millennium/Yang-Mills_Theory/:

The successful use of Yang-Mills theory to describe the strong interactions of elementary particles depends on a subtle quantum mechanical property called the "mass gap:" the quantum particles have positive masses, even though the classical waves travel at the speed of light. This property has been discovered by physicists from experiment and confirmed by computer simulations, but it still has not been understood from a theoretical point of view.

I learned that only particles with zero rest mass can move at the speed of light, so the sentence seems like a violation of special relativity. Also I don't understand what they mean by classical waves. Classically a particle with mass is a particle. There is no wave associated with it.

Can someone explain why this statement is not a violation of special relativity and what the Clay Mathematics Institute really means?