Sometimes the electrolyte directly takes part in the chemistry, by being oxidized or reduced at an electrode. In those cases, it is easy to understand why the concentration would have an influence on the kinetics of the redox reactions (and with that, on the observed electrical current).
For water electrolysis, the "active" electrolytes are hydrogen ions and hydroxide ions. If you change the pH, you change their concentration, and would expect to affect the rate of reaction. You are asking about other electrolytes such as sodium sulfate, which don't participate in electron transfer.
To explain their role, take a look at this animated GIF. It shows electrolysis of water in the presence of universal indicator and sodium sulfate:
With the 9 volts applied to the graphite electrodes, you can see that the electrolysis causes local changes in pH (because one half reaction consumes hydrogen ions and the other hydroxide ions). It takes quite a bit of effort to mix the solution (with the plastic tube) to get the pH homogeneous again. This shows you that the diffusion of hydrogen and hydroxide ions takes a while. In the absence of other electrolytes, this would lead to local charge accumulation as well, stopping the reaction. With the electrolyte (at much higher concentration than the hydrogen and hydroxide ion), we get a pH gradient only.
This pH gradient also slows things down, and increases the voltage that needs to be applied (cell potential is 1.3 V at constant pH, but according to Nernst law, it would increase because of the pH differences). Because we are applying 9 volts, this still works.
What an electrolyte “really” does in electrolysis?
By moving in the appropriate directions, the ions of the "passive" electrolyte prevent large accumulations of charge, and allow the electrolysis to proceed at higher rates than without their presence. If you decrease the gap between electrodes down to sub-Debye length, they are no longer neccessary.