Electrons as charge carriers

Electric charge is carried through wires and other conductors by electrons.
Electrons do not travel through solutions.

Ions as charge carriers

The moving particles in solutions are ions.
CATions travel to the CAThode. ANions travel to the ANode.
Anions being negatively charged deposit electrons on the anode.
Cations being positively charged pick up electrons from the cathode. Other electrons move round the circuit (from the anode) to replace them.
So in the circuit (and detecting devices like ammeters) electrons are flowing, just as if the entire circuit was made of conducting wires.
This explains how a solution conducts electricity.

How different factors affect current during electrolysis

The greater the concentration of ions the more are able to move through the solution in a given time interval.
The greater the voltage, the greater the charge on the electrodes so greater attraction for ions and greater current flow.
The greater the distance between electrodes means that the average time for ions to reach them is greater so the electron flow through the wires is less.