An ionic compound is a giant lattice of positive and negative ions. Every ion is held by strong electrostatic forces of attraction to the oppositely charged ions around it, and these forces act in all directions through the lattice.
To melt or boil an ionic compound, these strong forces between ions must be overcome. This needs a lot of energy, so ionic compounds have high melting points and high boiling points.
In a solid the ions are held in fixed positions in the lattice. They cannot move, so a solid ionic compound does not conduct electricity. When the compound is molten, or dissolved in water to make an aqueous solution, the ions are free to move. They carry electric charge through the liquid, so molten and aqueous ionic compounds conduct electricity.
