Take a simple salt such as $NaCl$. We might represent this as $Na^+Cl^-$. Now in the solid state, each cation ($Na^+$) is directly bound to 6 anions ($Cl^-$), and vice versa. There are secondary interactions throughout the lattice, and the solid state structure is highly ordered and stable. The individual ions are not free to move, and are held in an electrostatic lattice in fixed positions.
If the lattice is disrupted, say by supplying (a lot of!) heat to give molten $NaCl$, or more simply by dissolving the ionic solid in water, the individual ions are free to move, and can therefore transfer a current. Note that not all ionic solids are soluble in water; it depends on the strength of the lattice.
