The nuance is this: Naturally, in a field theory the word "particle" is ill-defined, thus the only true statement one can make is that: the propagator/green function of the field contains poles at +-m, which sort of hints at what he means by stiffness.
As a result of this pole, any perturbations of the field have an exponential decaying effect. But the pole is the mass, by definition.
The real interesting question is why Z and W bosons are massive, which have to do with the higgs mechanism. I.e., prior to symmetry breaking the fields are massless, but by interacting with the Higgs, the vacuum expectation value of the two point function of the field changes, thus granting it a mass.
In sum, whoever wrote this is a bit confused and just doesn't have a lot of exposure to QFT