In the bulk, the electric field moves ions, and the moving charge is a current.
In the wires, the electric field moves electrons.
At the contact between the conductor and the electrolyte, the ions and the electrodes exchange electrons.
In a first approximation, you can imagine that the dissolved ions get to the electrodes with which they exchange electrons, and that if the species created there are unstable in water (Na for instance) they react to release something else (H2 for instance).
Though, I have no hope that the detailed process is so simple. The reaction must happen while the species still stick at the electrode, through bizarre intermediates, and this must have consequences at the cell voltage and more. For instance, I expect no dissolved atomic hydrogen; it must stay adsorbed at the electrode until meeting an other hydrogen atom to form a hydrogen molecule.