This is really easy. To conduct electricity, you need FREELY MOVING particles with CHARGE. Not anything else, but moving particles with charge. In solid state between the particles of salt there are gaps, that you probably don't see, but electricity can't "jump", only if it has high current. So if it is melted, you have now moving particles with charge, that can conduct.
Metals with displaced sea of electrons because of the displacement can conduct electricity in solid state, because electrons have charge, relatively -1, and they move freely. Also, because of the big cloud electrons they tend to absorb light, or, reflect it in a strange way, this is why some of them are reflective, and most of them are greyish-black colored(absence of light is called black
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