In layman's terms, how do we know the speed of light cannot be exceeded?

Because we believe in causality, i.e. that cause always precedes effect. If it were possible to exceed the speed of light, then en event happening here and now could have affected something elsewhere before a light signal could reach that other point. Think about me being a spy on an enemy starship: i see that they are shooting a laser beam towards my friends, so I message them quickly "take cover, a laser has been shot at you". However, as special relativity tells us, the temporal ordering of events taking place outside each others light cone (that means, of events that can't be connected by light or slower signal) depends on the frame of reference. So in our hypothetical case some observers would have seen that the event that you "caused" by sending faster-than-light communication occured before I sent the signal: my friends run for cover, before the laser has been shot and before I have messaged them.

In response to your point (1) , the reason people believe relativity is correct has nothing to do with what we use to observe light with: we certainly don't use light to observe itself, rather we use the interactions of light with matter to observe the behavior of light. That the speed of light can't be exceeded is a consequence of the postulates of relativity, it means we assume that certain statements are true and then derive the consequences that follow. If the consequences of the postulates are observed to be false in the real world, then the theory will have to be discarded. The two postulates of special relativity can be stated many ways, here is one: (I) The laws of physics are the same in all inertial frames of reference. (II) The speed of light in a vacuum is the same in all inertial frames of reference. But relativity, believe it or not, does not depend on whether the speed of light is actually the fundamental speed limit of the universe. It could be that the particle of light itself is massive, and travels ever so slightly slower than the fundamental speed limit in the universe. No one believes this to be true, and all the evidence suggests that the photon is massless but it is not completely ruled out experimentally, that the photon has a very small mass. But relativity does absolutely say, that there is a fundamental speed limit in the universe that cannot be exceeded. This has very deep consequences for all of physics, when anything is moving at relative speeds that are close to that speed limit ... the kinetic energy of any object increases extremely rapidly as the speed limit is approached if the object is massive. This speed limit is routinely approached in particle accelerators all around the world. The accelerators wouldn't even work if the kinematics that are derived from the two postulates above were false, in a big way.

1. As others have already pointed out, we do not study light with light (there is no interaction of light with itself in vacuum). Others have already described relativity theory, so I will not expound on that. What needs to be said though is that in the late 19th century Michelson and Morley have attempted to measure the velocity of the Aether. In those days light was imagined to propagate through an invisible, all permeating "sea", like waves in the ocean. When Michelson and Morley measured it, they found the Aether to be stationary relative to all inertial frames. What that essentially means is that the speed of light is independent of the emitting device, observer, etc. This result alone would necessitate the speed of light to be the fastest possible (see the other answers for more details). 2. The Michelson–Morley experiment was conducted using interferometers, which are quite indifferent to the speed of light. (To measure anything faster you would just increase the path traveled through the device, using mirrors etc).   Thinking of photons as everyday particles is misleading.  As I remember one physics professor saying, "using the word photon should only be allowed to PhD's in physics". 3. The difference between sound and light is that sound has a propagating medium and light does not. (there are more differences which are irrelevant right now). A wave's frequency and wavelength are dependent on frame of reference. This is called the doppler effect. In the case of sound the change in frequency and wavelength are dependent upon observer's frame of reference, transmitter's frame of reference, both of which relative to the medium (air for example). In the case of light the only factor is the relative velocity between the receiver and transmitter.

Speed is a "velocity / time". Exceeding the speed of light would 'break' the time parameter of the equation. (That is a very simplified thought, in keeping with the request of the question.)

In Einstein's day, it was widely believed that the "Theory of Relativity" - (I think they were talking about Special Relativity) was so complex that only three people understood it. That's a gross exaggeration as many people do understand it today. You, however, are asking that it be explained in "laymen's terms". I bet the very question would have seemed preposterous back in the day. I found this definition for a layman : a nonordained member of a church. a person without professional or specialized knowledge in a particular subject. I'm sure it could be made accessible for some who are not members of a church. In fact, Einstein, himself, wasn't a church goer unless his occasional visit to a synagogue would count against him. It should also be accessible to a person without professional or specialized knowledge if that person is fairly logical and is motivated to focus hard. Others have already offered cogent explanations for what follows from the peculiar but oft observed fact that no matter where you are and no matter how fast you are moving, when you measure any electromagnetic wave (light being only a subset of electromagnetic emanations) its speed measures out to be always the same darn thing, a non varying, always predictable speed which we call "the speed of light". With that as a starting point, you can be lead through various thought exercises to establish that every other possible constant movement you are going to measure is always less than that special velocity. I warn you though, you have to get the dogs to stop that infernal barking while you concentrate.

This is the best explanation I found : Everything, by nature of simply existing, is "moving" at the speed of light (which really has nothing to do with light: more on that later). Yes, that does include you. Our understanding of the universe is that the way that we perceive space and time as separate things is, to be frank, wrong. They aren't separate: the universe is made of "spacetime," all one word. A year and a lightyear describe different things in our day to day lives, but from a physicist's point of view, they're actually the exact same thing (depending on what kind of physics you're doing). In our day to day lives, we define motion as a distance traveled over some amount of time. However, if distances and intervals of time are the exact same thing, that suddenly becomes completely meaningless. "I traveled one foot for every foot that I traveled" is an absolutely absurd statement! The way it works is that everything in the universe travels through spacetime at some speed which I'll call "c" for the sake of brevity. Remember, motion in spacetime is meaningless, so it makes sense that nothing could be "faster" or "slower" through spacetime than anything else. Everybody and everything travels at one foot per foot, that's just... how it works. Obviously, though, things do seem to have different speeds. The reason that happens is that time and space are orthogonal, which is sort of a fancy term for "at right angles to each other." North and east, for example, are orthogonal: you can travel as far as you want directly to the north, but it's not going to affect where you are in terms of east/west at all. Just like how you can travel north without traveling east, you can travel through time without it affecting where you are in space. Conversely, you can travel through space without it affecting where you are in time. You're (presumably) sitting in your chair right now, which means you're not traveling through space at all. Since you have to travel through spacetime at c (speed of light), though, that means all of your motion is through time. By the way, this is why time dilation happens: something that's moving very fast relative to you is moving through space, but since they can only travel through spacetime at c, they have to be moving more slowly through time to compensate (from your point of view). Light, on the other hand, doesn't travel through time at all. The reason it doesn't is somewhat complicated, but it has to do with the fact that it has no mass. Something that isn't moving that has mass can have energy: that's what E = mc^2 means. Light has no mass, but it does have energy. If we plug the mass of light into E=mc^2, we get 0, which makes no sense because light has energy. Hence, light can never be stationary. Not only that, but light can never be stationary from anybody's perspective. Since, like everything else, it travels at c through spacetime, that means all of its "spacetime speed" must be through space, and none of it is through time. So, light travels at c. Not at all by coincidence, you'll often hear c referred to as the "speed of light in a vacuum." Really, though, it's the speed that everything travels at, and it happens to be the speed that light travels through space at because it has no mass. This also covers the common question of why nothing can ever travel faster than light, and why things with mass cannot travel at the speed of light. Since everything moves through spacetime at c, nothing can ever exceed it (and no, traveling backwards in time would not fix that). Also, things with mass can always be "stationary" from someone's perspective (like their own), so they always have to move through time at least a little bit, meaning they can never travel through space as fast as light does. They'd have to travel through spacetime faster than c to do that, which, again, is not possible. Note :- I am not the author of this piece.I found it when I had the same question. Source :- http://www.reddit.com/r/explainlikeimfive/comments/22pi7o/eli5_why_does_light_travel/cgp58ml