How is saltwater able to conduct electric charge between two wires?
The salt (NaCL) water solution is an electrolyte solution which is, essentially, a conductive solution.

The conductivity of the salt water is due to the presence of both positively and negatively charged ions.  These ions in solution are free to accelerate in the presence of an electric field and thus, like the free electrons in a metal conductor, are able to participate in an electric current (not to be confused with an electron current).

When there is an electric current through the salt water, there are actually two contributions:  (1) the positive sodium ions drifting in the direction of the electric current and (2) the negative chlorine ions drifting the opposite direction.

While it may seem that the oppositely directed ion currents should cancel they, in fact, add.  The flow of negative ions contributes to an electric current in the opposite direction due to the negative sign of their charge.

At the interface between the metal conductor and salt water, there are reactions that either remove electrons or add electrons to the conductors thus completing the path for charge to flow around the circuit.

From the online GenChem Textbook section on Electrolysis:




  Figure 1 An electrolytic cell. The battery pumps electrons away from
  the anode (making it positive) and into the cathode (making it
  negative). The positive anode attracts anions toward it, while the
  negative cathode attracts cations toward it. Electrical current is
  carried by electrons in the wire and electrodes, but it is carried by
  anions and cations moving in opposite directions in the cell itself.
  Since the anode can accept electrons, oxidation occurs at that
  electrode. The cathode is an electron donor and can cause reduction to
  occur.


It is important to note that electric current is simply defined as the flow of electric charge and this definition does not depend on the species of charge carrier.

There is, in fact, just one electric current in the circuit while there are three (or more) different species of charge carriers along the circuit's path:  (1) electrons in the metal conductors, (2) positive sodium ions in the salt water, (3) negative chlorine ions in the salt water, and (4+) if the voltage source is a battery, ions in the battery's electrolyte.