How do we know that light cannot travel faster than it does? We assume the speed of light in vacuum is its maximum speed but can we not assume that it could be faster, or slower?
 A: Light can be treated as an electromagnetic wave. Electric and magnetic fields have some properties: if electric field changes the magnetic field appears around, if magnetic field changes - electric field appears. There are accurate mathematical formulas which describe these laws (Maxwell's laws). There are some constants in these formulas which depend on the properties of surrounding substance. A consequence of these mathematical formulas is that there electric and magnetic field can exist without charges and propagate with some constant speed that depends on the constants in formulas.
If properties of surrounding substance are different, constants in Maxwell's laws are different, electromagnetic waves propagate with different speed.
Properties of vacuum are the same everywhere. (Why - it's a different question, I do not know). So, the speed of light in vacuum is constant everywhere.
In some substance light can propagate slower than speed of light in vacuum $c$. But not faster. It's a consequence of special relativity theory.
Special relativity theory theory states that time and space are not independent. If there are two events it's not always possible to tell which event happened earlier. It may happen that in one frame of reference the first event happened earlier, in some other frame of reference - the other one happened earlier. The formulas which translates time and coordinates of some event in one frame of reference to time and coordinates in another frame of reference contain some constant speed $c$. Would it be possible to send information faster than $c$ it would be also possible to transmit information between several stations A -> B -> C -> A so that A receives the information before it was sent. Time machines like this would bring miscellaneous paradoxes, so moving faster than that speed $c$ from special relativity formulas is not possible. 
Why the speed of light in vacuum and the constant $c$ from the special relativity formulas are the same thing - it's yet another separate question.
A: We used to think it could. Experiments proved that wrong.
In the 19th century, it was widely believed that light, being wave-like, must be propagating through a medium that permeates the universe, just like sound waves propagate through air. This hypothetical medium was called "luminiferous aether".
Many experiments were conducted to prove the existence of aether, and in particular to answer an important question: is aether stationary, or is it dragged along by moving matter? The most famous of these experiments was the Michelson-Morley experiment, which attempted to detect the Earth's motion through stationary aether, by comparing the speed of light in beams parallel and perpendicular to this motion.
However, the experiment did not produce the expected result, and in fact all the experiments on aether led to an odd conclusion: the speed of light seemed to always be the same, regardless of the motion of the light source, the observer, and the path taken by the light. In time, this led to Einstein formulating his theory of relativity, and the aether theory being abandoned.
