Why does higher temperature affect the rate of electrolysis?
Rate of electrolysis is only dependent on the current (Faraday's law) passing through the solution. Remember that in an electrochemical experiment you cannot independently vary voltage or current. One factor has to be fixed. These two modes are called constant current electrolysis or constant potential electrolysis.
Once you have fixed voltage, you can ask you yourself, what factors affect current rather than electrolysis. The key factor is solution's resistance or better conductance (reciprocal of resistance). Conductance increases with temperature.