Although solid ionic compounds are made up of ions but they do not conduct electricity in solid state. This is because in the solid ionic compound the ions are held together in fixed positions by strong electrostatic forces and cannot move freely. However, when we dissolve the ionic solid in water or melt it, the crystal structure is broken down and ions become free to move and conduct electricity.
Thus, an aqueous solution of an ionic compound conducts electricity because there are plenty of free ions in the solution which are able to conduct electric current.