The reason matter cannot reach the speed of light is that theres no way to push it that fast. Simple, eh?
Going into the mathematics a bit further, in order to move something you must apply force(F). If you have taken a physics course you probably remember that
F = m*a
or force used is equal to mass accelerated times the acceleration experienced. And this is true, up to a point. What force actually is defined as, however, is the change in momentum with respect to time. In calculus this is expressed as
momentum = m*v
F = d(m*v)/dt
In classical mechanics this equation is equivalent to F=m*a, because the derivative of velocity(v) with respect to time is acceleration(a), and mass(m) is a constant with respect to time. However, because of relativity effects, mass is dependent on velocity according to the equation
m = m0 / sqrt(1 - v^2/c^2)
where m0 is the rest mass of the object, v is the current velocity, and c is the velocity of light in a vacuum. From this it can be seen that as velocity increases the force needed to accelerate the particle also increases. Furthermore, as the velocity approaches c the force required approaches infinity. This means that in order to accelerate an object to lightspeed you would require infinite force; because an infinite force is impossible, it is impossible for matter to travel at the speed of light.