Why soak filter paper in K2SO4?
For this experiment, you'll be seeing an interesting color change when Iodine is oxidized at the anode to form the brownish $\ce{I2}$ on the electrode and yellow $\ce{I3-}$ in solution.

The reduction of $\ce{H+}$ to form $\ce{H2~(g)}$ and $\ce{OH-}$ will turn the indicator pink at the cathode. 

The cool thing about your experiment is that you will not see any redox reaction at all (at either electrode) until iodide ions diffuse to the anode! That is, unless you use a high voltage battery, $\gtrapprox2.0~\mathrm{V}$, you will not have sufficient energy to oxidize $\ce{SO4^2-}$ to $\ce{S2O8^2-}$. 

This makes the experiment an exercise in patience, and then sudden color display! Measuring the time it takes can yield data that can help you calculate the diffusion coefficient of $\ce{I-}$ if you wanted to take it another step!

In summary: Soaking the filter paper with KI would result in the electrodes producing color changes immediately. I suspect the sulfate was used in this experiment to illustrate the diffusion of dissolved ions, because the redox reaction would not take place until the iodide ions reached the anode.