To answer this question we must recall that electrical conductivity is a measure of how many charges are moving, and how fast. For HCL, the answer comes down to how fast the charge carriers can move.
For HCL the ionization constant is very high, which means that for reasonably dilute solutions, HCL is completely dissociated. So there are a lot of charge carriers, which is necessary for high conductivity. When HCL dissociates in H2O, the H+ proton diffuses to the free electron pair in H2O and becomes a H3O+ cation. In the presence of an electric field, the H+ proton will "hop" from water molecle to water molecule, much like an electron would in a metal.
In anhydrous HCL, there is no such "hopping"; the charge carrier is not just the H+ proton, but the entire solvent molecule, which must be physically transported by the electric field. This is a much slower effect.