Let me try without showing any equation :)

We start with the Lagrangian describing the dynamics of the particle physics model. This Lagrangian is an equation containing everything (which particles we have, how they interact, what are their masses, etc.). The Lagrangian must include in particular a mass term for each of the massive particles. However, writing such a mass term would directly violate the symmetry principle you imposed to start with. So mass terms (and thus particle masses) are forbidden. I hope it helps (a little at least) :)