Well, the mathematics does not tackle physical mechanism generally. It only gives quantitative analysis.
There is no physical mechanism that prevents accelerating faster than light.
What limits the acceleration within speed of light, is speed with which of the accelerating force itself acts. Whenever we accelerate something, we (in most cases) accelerate by applying electromagnetic force (repulsion). When you push a car with your hand, the electrons in your hand repel the electrons of the car at the point of contact.
Now considering the electromagnetic repulsion as cause of acceleration, the acceleration will be limited to the speed with which the force propagates. The force propagates at speed of light. So, it can not accelerate anything faster than it itself can act.
Imagine a car passes by you at 300 miles/hour. Do you think you can accelerate it further by pusing with your hands. Most likely not. Because human beings can not move their hands faster than 300 miles/hour, not even close.
The electromagnetic repulsion propagates through space to act on anything. That propagation speed is c. Something already moving at c, the force does nothing in that direction.
Therefore, as a body approached c, any force you apply on it, becomes ineffective and so, it does not accelerate further in that direction.
However, the force is effective if you want to slow it down because, then it has to act in opposite direction and it is able to.
Other type of forces also act at c. So in general it is not possible for the forces to accelerate anything faster than c.
The natural next question is why the forces propagate at c. Well that is a property, not a limit. It is property because the forces propagate exacly at c, no faster, no slower. It would have been a limit if the forces propagated at various speeds up to c. But they do not.
Edit: Infinite mass argument - Lot of people are giving argument that the mass approaches infinity as speed approaches c.
I am sure this also fits into mathematics, otherwise people would not be making this argument.
I may be wrong, and please feel free to correct me if you think so. But I do not think that is the case - i.e. mass does not approach infinite.
My simple argument is - if the mass of a moving particle approaches infinite and that infinite mass moving at speeds close to c, then it would be almost impossible to stop that particle.
We all know that though it is not possible to accelerate the particle further, it is no big deal to slow it down. Slowing down an infinite mass would not be that easy. Infinite mass reasoning must apply both ways - in speeding up as well as in slowing down. Which it does not seem to.
Therefore, I can argue that mass does not approach infinite, it is the forces that are rendered ineffective at such speeds.
This may be an example where it makes mathematical sense, but no real sense. Please correct if I am missing something, instead of blank down voting.