Dielectric constant is not an indicator of being a good insulator. The electrical conductivity is what gives a material it's properties relating to being an insulator or conductor. Take a look at air, it has a dielectric constant of unity, yet it is a very good insulator.
A high dielectric constant will result in a slower propagation of light through the medium and it allows for polarization of the medium. The polarization is the ability for the molecules or atoms in the medium to slightly separate to become dipole moments in the presence of an applied electric field. This has the effect of lowering the apparent magnitude of the electric field, but the energy lost in the field is stored in the polarization of the material. This helps make a dielectric effective in improving capacitors since you can store energy in the polarizing of the dielectric as well as in the voltage induced by the stored charges.