The point is that electromagnetic radiation itself has a velocity, namely the speed of light in the medium it's travelling in. Since energy can be transferred to an electron, presumably, only when a wave of energy catches up with it, obviously it's impossible for the electron to ever reach the speed of the wave influencing it. If you can't see this immediately, consider these everyday models of the situation, which I've tried to make as varied as possible to get the point across.
A floating toy boat which is pushed
Along by the waves. However big the waves are, the boat won't go faster than these waves.
Or consider a boy throwing stones, every second, at the same speed, at a floating piece of wood; however heavy the stones, the piece of wood will never b2c datasets travel faster than the stones. (Or at any rate once it travels faster than the stones, the stones can't catch up with it). But nobody would imagine the piece of wood must be getting heavier as it picks up speed, because it moves less when it's hit.
Or imagine a children's roundabout
The sort turned by hand. If an adult regularly swings his arm to turn the roundabout, as it approaches the rate the arm's swinging at, will never speed up beyond the velocity of the arm. It seems physicists, looking at electrons and measuring their speeds as they vary with energy, ignore this simple fact. They interpret the result as the particle getting heavier, with limiting speed that of light, without realising that the limit is imposed by their equipment. They assume in one part of their minds that electromagnetism travels at infinite speed.