I'm somewhat confused about this aspect of Special Relativity. I'm trying to think of this in terms of what an observer would see if they were a particle in a particle accelerator, and then what the person operating the accelerator (stationary with respect to the accelerator, but not the particle) would see.
Case 1: Observer who is an accelerating particle.
Since, from your perspective, the particle accelerator is moving and you yourself are stationary, the particle accelerator appears to be time dilated. The amount of energy you experience its electromagnets to be generating must also appear to be less, then. As you approach C, the amount of energy would begin to seem negligibly small. That seems to explain why it takes greater and greater amounts of effort to accelerate particles as they approach the speed of light.
Case 2: Observer who is operating the accelerator
I guess you perceive the mass of the particle to be increasing, because it's harder and harder to increase its speed by using your electromagnets to accelerate it?