This equation:
...is derived from relatavistic mathematics, derived from the observation that the speed of light is constant regardless of one's reference frame.
That is, if you throw a baseball forward from a moving train, the speed of the baseball as measured by an observer on the ground will be the speed of your pitch plus the speed of the train. If you throw it towards the rear of the train, the ball will be seen by the outside observer as travelling at the speed of the train minus the speed of your pitch.
Light, however, doesn't behave that way - the observer on the ground will measure the speed of the "pitched" beam of light to be exactly "c", regardless of which direction it's sent. This was confirmed by experiments by Michaelson & Morley at the US Naval Academy.
This leads to the odd results described by this equation and others as you approach the speed of light, or encounter a very strong gravitational field.
There's physical evidence of time dilation - there's a certain type of subatomic particle, called a "mu meson," that is created as cosmic rays impact the upper atmosphere, that has a very definite, short lifetime.
Based on a simple calculation of the speed at which these particles are travelling when they are created in the upper atmosphere, the laboratory-observed lifetime, and the distance from the upper atmosphere to the surface, they should hardly ever reach the surface of the Earth, because the distance is apparently too far to traverse before they decay.
But they do reach the surface and are detected, and this is due to their tremendous speed approaching the speed of light - their lifetime is extended due to time dilation. While one nanosecond passes for the particle, nine nanoseconds pass on the Earth, and through that, enough time elapses for the particle to reach the surface before its decay.