#1. The universe has no edge, no center, i.e., that no matter where in the universe you are, it stretches out in all directions as if you were at the center.
That's it. That's all insight you need to understand the theories. Everything else follows from it.
From #1 follows:
#2. The position of any object in the universe can only be defined in terms of other objects in the universe. For example, the position of the earth is generally defined relative to the position of the sun. "Absolute" positions (i.e., not defined in terms of other objects) do not exist.
#3. Since the position of objects can only be defined in terms of positions of other objects, this automatically also holds true for velocity. The speed of an object can only be defined in terms of speeds of other objects. For example, the speed of the sun in our solar system is (close to) zero (by definition), but generally non-zero relative to other stars. Any non-accelerating object may equally well be viewed as being stationary. There are no "absolute" velocities in the universe, since measuring an absolute velocity would require a stationary object holding a fixed absolute position in space, but we said absolute positions do not exist (#2).
#4. The speed of light traveling through space is constant.
Now imagine a non-relativistic universe. Then, #4 would contradict #3 (and therefore #2 and #1). Since if the speed of light is constant, an observer standing on some rock in space could measure its absolute velocity in the universe by measuring up how fast photons pass it by. If the observer finds that the speed of photons coming from some direction is 99% of c, then the observer would rightfully conclude that his rock was moving at 1% of c in that same direction.
Einstein understood that "position is relative" and "speed of light is constant" were both true. But that means that it must be impossible for an observer to measure his speed relative to the speed of light:
Imagine an observer in a spaceship who wants to establish its absolute speed in the universe. He switches off all engines and measures the speed of light in all directions and finds it to be exactly c. Not knowing the universe is relativistic, he concludes he is exactly stationary. Next, he speeds to 10% of c in some direction, switches off his engines and again measures the speed of c. To his surprise, he again finds the speed of light is c in all directions!
No matter how fast the observer moves (relative to its original speed), he always measures the speed of light to be c in all directions. The observer always sees photons pass him at a speed of c. Even when travelling at 99.9999% of the speed of light relative to a photon source, he still sees these photons passing him by at the speed of c.
The observer establishes the velocity of a photon by is measuring how much time it took the photon to travel from A to B. If the speed of c is constant, and at the same time the observer always measures c regardless of his own velocity, this must mean that clocks and dimensions of his spaceship must vary.
For instance, when moving away from a planet at 99% of c, photons coming from that planet are still being measured to have a speed of c. The time a photon coming from the planet takes to travel some fixed distance is constant regardless of the speed of the spaceship relative to the planet. This means that clocks on board of the spaceship must be moving slower than clocks on that planet, and such that the time the photon takes to travel a fixed distance, is fixed and c for the observer.
See also http://en.wikipedia.org/wiki/Consequences_of_special_relativity