Why can nothing travel faster than the speed of light?
The speed of light (in vacuum) is not like any other "speed". It is a basic constant of nature, which seem to be imprinted into the structure of space - time. In a basic sense, which I will explain immediately, it is equivalent to zero time. Thus, traveling faster than light would imply "reversing time", which would violate causality: namely, if you could travel faster than light, you could be born before your parents...
It is an experimental fact, first discovered by Michelson and Morley, that the speed of light is constant for any observer. Even for two observers which are moving with respect to each other (say, one at home and one inside a moving train) - still, both will measure the same speed for light.
In order to explain this counter-intuitive result, Einstein came up with the theory of relativity. Since speed is constant, the only way to reconcile this fact is to deduce that space and time (velocity = distance [space] / time) are not independent, but connected into one space-time. This means that two different observers that are moving with respect to each other, each measures different times on his / her clock; and both are right !.
Now, when someone starts traveling at faster and faster speeds, the time it takes this person to get from point A to point B, as measured by his own clock, gets shorter and shorter. If this person could travel at the speed of light (which he can never reach!), his time would be exactly zero !. Indeed, when we look at a photon (that travels exactly at the speed of light) that was emitted, say, by a star 10 light years away (1 light year is a measure of distance = the distance a photon travels in 1 year), we would say that it took the photon 10 years to reach us. However, if we could ask the photon, from the photon’s perspective, it took zero time from emission to detection, regardless of the distance it travelled.
An object with a mass can, in fact, never reach the speed of light. If we accelerate it, we would increase its speed. However, as its speed gets closer and closer to the speed of light, its energy increases even faster (we can think of it as the particle becomes heavier and heavier), and so to get to the speed of light it takes infinite amount of energy. Every finite amount of energy could accelerate massive particles to speeds close to that of light, but not exactly equal to it. Furthermore, as explained above, the closer the particle's speed is to the speed of light, less and less time it takes it to reach from point A to point B.
Now the sentence at the beginning of this answer makes more sense (I hope): since, from the perspective of a photon that travels at the speed of light, it takes zero time to travel between any two points in the universe. Thus, if anything could travel faster than light, it would take negative time to travel from A to B, and thus it will reach point B before being at point A... this makes no physical sense.
I should emphasise that although it was proposed that particles that travel faster than light (known as "Tachyons") may exist, there isn't any observational evidence - direct or indirect for such particles.