Basically, the universe has a speed limit. No object can ever exceed the speed of light.
Now imagine you decide to prove Einstein wrong by building a train capable of nearly reaching the speed of light, and then shooting a bullet forward in that train, so that the bullet will break the speed of light. In order to preserve this speed limit, time for you and your bullet, as determined by relatively moving observers, will be slower inside this train.
Whereas you will see the bullet shooting at normal speed, an observer outside the train, in the brief instant the train is in view, would see both you and the bullet moving in slow motion. Hence, relative to the train, and relative to the outside observer, the bullet will never actually reach the speed of light.
Note that speed is the relationship between two quantities: distance and time (i.e. meters per second or miles per hour). Hence, in order to maintain that speed limit, as an object moves faster through space, motion through time is slowed in the frame of reference of that object. The length of the object is, because of speed being the relationship between distance and time, also decreased, relative to other objects.
The change in time and length also affect other quantities, such as mass, force, and energy.
It's a little hard to believe, but in every single experiment ever used to test this theory, it has held true.