From I = V/R if you increase the voltage you increase the current, so you get a greater amount of charge passing a point at a particular time. Does this mean the electrons travel faster or you get a greater quantity of electrons passing the point?
You will find it is a little bit of both, if you imagine you had a hose pipe, and turned the tap on halfway, you would get a steady stream of water from the other end, now if you turn the tap all the way up, you get much more water coming out of the end with a greater pressure, so because of the pressure increase and the resistance of the walls of the hose, there would be more water molecules in the same space, likewise the molecules would move faster because of the build up of pressure behind them.