Why does painting a silvered radiator with black paint improve it's ability to radiate heat?

Because radiated heat is heat transferred by light (that is, radiation). The paint is in physical contact with the radiator, so the paint gets heating by conduction, and the paint then cools by radiating heat (as well as a little conduction with the air, and a lot of convection).

Unless the radiator is getting hot enough to glow in optical wavelengths the optical color doesn't matter. If it is hot enough to glow, then black will perform better than white because the ability to emit radiation is the same, from a physics perspective, as the ability to absorb it. If the radiator isn't that hot, then what matters is the "color" in wavelengths you can't see, and shiny metal is good at reflecting light in many wavelengths, so is bad at absorbing/emitting light.

Completing the other answer, radiator color only has a direct impact on the emitted wavelengths of the radiator. A radiator at 373K (100 C / 212F) radiates most energy in the Mid-wavelength infrared (3–8 µm) range, outside the visible spectrum.

The only significant feature of the radiator coat at it's radiating wavelength is transparency. The less it absorbs or reflects light, the larger the effective radiating layer.

As an approximation, only the molecules inside the radiative penetration depth region of the material have photons which will reach the exterior of the material, instead of being absorbed inside. So for a given temperature the amount of heat radiated will be approximately proportional to penetration depth at the radiated frequency.