Ionic compounds can conduct electricity when dissovled in water or mellted, becasue

Because the ions forming the compound will have to separate from each other then. As a result, ionic compounds disintigrated into a myriad of single lonely ions floating free in water (in solutions). The process is known as electrolytic dissociation.

Eletric current is co-directed movement of particles exhibiting similar charge.

Water doesn't have charged particles constituting it. No electricity.

Ionic compounds do have charged particles (i. e. ions), but those are locked within the lattice and cannot move. No electricity.

Ionic compounds in water disintegrate into free floating (capable of moving) ions (charged particles). All you have to do is get all the positive charges moving one way and all the negative ones moving the other and here you have it - electricity! That is achieved through applying external magnetic field.

This is why ionic compunds can conduct electricity when dissolved in water or melted.