There are a lot of simple, intuitive explanations of this to be had out there … but I kind of hate them all. You might google around a bit and find discussion of something called "relativistic mass," and how it requires more force to accelerate an object that's already moving at a high velocity, stuff like that. That's a venerable way of interpreting the mathematics of special relativity, but I find it unnecessarily misleading, and confusing to the student who's just dipping her first toe into the ocean of modern physics. It makes the universe sound like a much different, and much less wonderful, place than it really is, and for that I kind of resent it.
When I talk about this subject, I do it in terms of the geometric interpretation that's consistent with general relativity. It's less straightforward, but it doesn't involve anything fundamentally more difficult than arrows on pieces of paper, and I think it offers a much better understanding of the universe we live in than hiding behind abstractions like "force" and outright falsehoods like "relativistic mass." Maybe it'll work for you, maybe it won't, but here it is in any case.
First, let's talk about directions, just to get ourselves oriented. "Downward" is a direction. It's defined as the direction in which things fall when you drop them. "Upward" is also a direction; it's the opposite of downward. If you have a compass handy, we can define additional directions: northward, southward, eastward and westward. These directions are all defined in terms of something — something that we in the business would call an "orthonormal basis" — but let's forget that right now. Let's pretend these six directions are absolute, because for what we're about to do, they might as well be.
I'm going to ask you now to imagine two more directions: futureward and pastward. You can't point in those directions, obviously, but it shouldn't be too hard for you to understand them intuitively. Futureward is the direction in which tomorrow lies; pastward is the direction in which yesterday lies.
These eight directions together — upward, downward, northward, southward, eastward, westward, pastward, futureward — describe the fundamental geometry of the universe. Each pair of directions we can call a "dimension," so the universe we live in is four-dimensional. Another term for this four-dimensional way of thinking about the universe is "spacetime." I'll try to avoid using that word whenever necessary, but if I slip up, just remember that in this context "spacetime" basically means "the universe."
So that's the stage. Now let's consider the players.
You, sitting there right now, are in motion. It doesn't feel like you're moving. It feels like you're at rest. But that's only because everything around you is also in motion. No, I'm not talking about the fact that the Earth is spinning or that our sun is moving through the galaxy and dragging us along with it. Those things are true, but we're ignoring that kind of stuff right now. The motion I'm referring to is motion in the futureward direction.
Imagine you're in a train car, and the shades are pulled over the windows. You can't see outside, and let's further imagine (just for sake of argument) that the rails are so flawless and the wheels so perfect that you can't feel it at all when the train is in motion. So just sitting there, you can't tell whether you're moving or not. If you looked out the window you could tell — you'd either see the landscape sitting still, or rolling past you. But with the shades drawn over the windows, that's not an option, so you really just can't tell whether or not you're in motion.
But there is one way to know, conclusively, whether you're moving. That's just to sit there patiently and wait. If the train's sitting at the station, nothing will happen. But if it's moving, then sooner or later you're going to arrive at the next station.
In this metaphor, the train car is everything that you can see around you in the universe — your house, your pet hedgehog Jeremy, the most distant stars in the sky, all of it. And the "next station" is tomorrow.
Just sitting there, it doesn't feel like you're moving. It feels like you're sitting still. But if you sit there and do nothing, you will inevitably arrive at tomorrow.
That's what it means to be in motion in the futureward direction. You, and everything around you, is currently moving in the futureward direction, toward tomorrow. You can't feel it, but if you just sit and wait for a bit, you'll know that it's true.
So far, I think this has all been pretty easy to visualize. A little challenging maybe; it might not be intuitive to think of time as a direction and yourself as moving through it. But I don't think any of this has been too difficult so far.
Well, that's about to change. Because I'm going to have to ask you to exercise your imagination a bit from this point on.
Imagine you're driving in your car when something terrible happens: the brakes fail. By a bizarre coincidence, at the exact same moment your throttle and gearshift lever both get stuck. You can neither speed up nor slow down. The only thing that works is the steering wheel. You can turn, changing your direction, but you can't change your speed at all.
Of course, the first thing you do is turn toward the softest thing you can see in an effort to stop the car. But let's ignore that right now. Let's just focus on the peculiar characteristics of your malfunctioning car. You can change your direction, but you cannot change your speed.
That's how it is to move through our universe. You've got a steering wheel, but no throttle. When you sit there at apparent rest, you're really careening toward the future at top speed. But when you get up to put the kettle on, you change your direction of motion through spacetime, but not your speed of motion through spacetime. So as you move through space a bit more quickly, you find yourself moving through time a bit more slowly.
You can visualize this by imagining a pair of axes drawn on a sheet of paper. The axis that runs up and down is the time axis, and the upward direction points toward the future. The horizontal axis represents space. We're only considering one dimension of space, because a piece of paper only has two dimensions total and we're all out, but just bear in mind that the basic idea applies to all three dimensions of space.
Draw an arrow starting at the origin, where the axes cross, pointing upward along the vertical axis. It doesn't matter how long the arrow is; just know that it can be only one length. This arrow, which right now points toward the future, represents a quantity physicists call four-velocity. It's your velocity through spacetime. Right now, it shows you not moving in space at all, so it's pointing straight in the futureward direction.
If you want to move through space — say, to the right along the horizontal axis — you need to change your four-velocity to include some horizontal component. That is, you need to rotate the arrow. But as you do, notice that the arrow now points less in the futureward direction — upward along the vertical axis — than it did before. You're now moving through space, as evidenced by the fact that your four-velocity now has a space component, but you have to give up some of your motion toward the future, since the four-velocity arrow can only rotate and never stretch or shrink.
This is the origin of the famous "time dilation" effect everybody talks about when they discuss special relativity. If you're moving through space, then you're not moving through time as fast as you would be if you were sitting still. Your clock will tick slower than the clock of a person who isn't moving.
This also explains why the phrase "faster than light" has no meaning in our universe. See, what happens if you want to move through space as fast as possible? Well, obviously you rotate the arrow — your four-velocity — until it points straight along the horizontal axis. But wait. The arrow cannot stretch, remember. It can only rotate. So you've increased your velocity through space as far as it can go. There's no way to go faster through space. There's no rotation you can apply to that arrow to make it point more in the horizontal direction. It's pointing as horizontally as it can. It isn't even really meaningful to think about something as being "more horizontal than horizontal." Viewed in this light, the whole idea seems rather silly. Either the arrow points straight to the right or it doesn't, and once it does, it can't be made to point any straighter. It's as straight as it can ever be.
That's why nothing in our universe can go faster than light. Because the phrase "faster than light," in our universe, is exactly equivalent to the phrase "straighter than straight," or "more horizontal than horizontal." It doesn't mean anything.
Now, there are some mysteries here. Why can four-velocity vectors only rotate, and never stretch or shrink? There is an answer to that question, and it has to do with the invariance of the speed of light. But I've rambled on quite enough here, and so I think we'll save that for another time. For right now, if you just believe that four-velocities can never stretch or shrink because that's just the way it is, then you'll only be slightly less informed on the subject than the most brilliant physicists who've ever lived.
EDIT: There's some discussion below that goes into greater detail about the geometry of spacetime. The simplified model I described here talked of circles and Euclidean rotations. In real life, the geometry of spacetime is Minkowskian, and rotations are hyperbolic. I chose to gloss over that detail so as not to make a challenging concept even harder to visualize, but as others have pointed out, I may have done a disservice by failing to mention what I was simplifying. Please read the follow-ups.
Well in a sense the poster doesn't really explain why FTL travel cannot exist, merely that light travels as fast as it can within the limits of the universe. The limit in the above example being the length of the arrow. Why that limit exists is a totally different question and as of now no-one knows the answer.