We have a natural feel for time space. We live in a three dimensional universe, where the length of an object is unchanging, where time passes in the same way for every object, and where the velocity of a non-accelerating object changes by the amount that my own velocity changes. If I measure the length of a car parked at the curb, I will not see this value change when the car is driven away. If I set two accurate watches to the same time, I expect them to remain in synchronization regardless of where I place them. If I throw a ball from a moving car to a bystander, the bystander will see the ball moving at a velocity equal to its velocity when leaving my arm plus the velocity of the car. This is the universe that we see and are physiologically designed to understand.
Special relativity overturns this natural picture of space and time. For those not accustomed to approaching a problem with a fresh mind, which usually includes everyone studying modern physics for the first time, special relativity is a shock, because it is so contrary to our personal experiences. The point, however, is that our personal experiences are limited, and to solve any problem in physics, a scientist must understand where these limits lie.
Special relativity was developed to solve a serious problem in the theory of electromagnetism. In the 19th century, physicists developed a set of four equations, named Maxwell's equations, that describe how charge and current create electric and magnetic fields and how these fields in turn exert a force on charged bodies. These equations are purely empirical; they are derived from experiments with electricity. But to accept these equations as they are contradicts our daily experiences with how object move, because the equations produced an equation for electromagnetic radiation that admitted only one speed, the speed of light.
In the 19th century, few scientists were able to look beyond their own experiences of how the speed of an object changes when an observer's own velocity changes. The common reaction to the problem of only one speed was to assume that Maxwell's equations are incomplete. More precisely, because the equation for electromagnetic radiation is a wave equation, scientists of that era thought that terms describing the velocity of a material through which the waves propagated was missing. These scientists has as an example the propagation of sound through air, and so they naturally extended this understanding of sound waves to electromagnetic waves, and assumed that there must be an as-yet undetected aether that serves as a carrier for electromagnetic waves. The speed of the waves relative to the aether would be the light speed, and the velocity of the waves for an observer would be the light velocity plus the aether velocity.
Einstein's genius was that he could accept the Maxwell equations at face value. The focus of the problem then shifts to how our motion affects the observed velocity of an object when we require that our motion does not change the form of Maxwell's equations. The problem therefore becomes a straight-forward mathematical problem of constructing a set of equations for transforming an object's velocity when an observer changes his own velocity. This set of equations and the equations of motion consistent with these transforms are the equations of special relativity.
The nature of the changes we must make to our intuitive understanding of space and time is easily understood by thinking about the consequences of the speed of light remaining unchanged as I change my motion.
If light propagated through an aether, and if I were not moving relative to that aether, light would travel at a single velocity regardless of direction. But if I begin to move in one direction relative to the aether, the velocity of the light would change by an amount opposite to my change in velocity. So light traveling in the direction of my motion would be moving at the light speed relative to the aether minus my speed, while the light traveling opposite to my direction of motion would be moving at the light speed through the aether plus my speed. The wavelength of a periodic wave is unaffected by my motion, because measurements of length are unaffected by motion. The frequency of the wave, however, does change, because a periodic wave must travel a different distance when I am moving than when I am at rest for the periodic pattern to complete a full cycle at my position. So the wave traveling in my direction must travel a longer distance through the aether for it to complete a periodic cycle along side of me than if I were not moving, so I would measure a longer time between complete periods, and a longer frequency. A wave moving in the opposite direction than myself travels a shorter distance to complete a cycle, since my motion is in part closing this distance, so the time of a complete cycle becomes shorter, and the frequency becomes longer. This was the behavior that the physicists of the 19th century were attempting to impose on Maxwell's equations.
If I accept that Maxwell's equations are complete, so that the hypothesize of an aether is superfluous, I must change the description of the last paragraph of how the velocities of other objects change when I change my own velocity. My starting position is by definition the rest position—I no longer have an aether to which I can compare my motion, so I must arbitrarily choose what constitutes being at rest; I choose my desk. In this “inertial reference frame,” I will choose a set of sources for the electromagnetic waves, so I choose a pair of radio broadcast towers, each transmitting at the same frequency for anyone sitting still like I am, one directly to my east, and the other directly to my west. The radio waves coming from both towers are traveling with the same speed, frequency, and wavelength. If I begin moving west, I still find that the speed of light is the same for the waves from either tower, and this value equals the value I found when sitting at my desk; my velocity has no effect on the speed of the waves. To preserve a consistent picture, where the frequency times the wavelength equals the speed, my measurement of each wave's wavelength and frequency must change from what is found at rest. For instance, if the frequency increases as I accelerate from a rest position, I will measure a shorter wavelength for the wave. This illustrates the central feature of special relativity: my measurements of time and length must depend on my motion if the speed of light is to be constant as I accelerate.