Why do ionic compounds conduct electricity only when molten or in an aqueous solution?

Electricity is conducted when charges move from one place to another. In metals, the conductivity is due to delocalized electrons that can wander all over the place. You're probably already familiar with this.

In solid , on the other hand, the ions aren't going anywhere. The cations and anions all attract themselves into a nice lattice and they're stuck there until they have a better reason to wander off. No moving ions = no electricity.

However, if you melt an ionic compound or dissolve it in water, these ions are now free to move around. As a result, the solution can be made to conduct electricity.

Think about this, in solid form the outer shells of the elements are filling each other thus neutralizing the substance, but when you separate the ions (atoms of each element) their shells have space that needs to be filled or emptied, so they want to gain or loose electrons to have a full energy level. Because they want this the electrons are able to move and "bounce" from ion to ion.