The passage of electricity through an electrolyte solution is caused by

The Electrolyte when dissolved in polar solvent like H₂O dissociates into positive specie called Cation and a negative specie called Anion. When an electric current is applied to Electrodes dipped in solution of electrolyte, the cations move toward the negative electrode while anions move towards positive electrode. At electrodes the exchange of electrons take place.

Result:

So, the passage of current in electrolyte solution is due to both Cations and Anions.