In order for something to conduct electricity, it must have delocalised (freely moving) charge carriers, which in materials will be electrons, since electricity is the movement of charge. Insulators don't have freely moving electrons as they are all localised on atomic nuclei. However, in conducting materials like metals, there are delocalised electrons which can freely move, and so when a voltage is induced, the electrons move and so current is produced, and electricity is conducted. I hope this helps.
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