I'm not trying to disprove it, I am trying to understand it. <_<
You need to kinda toss out what you know about time to understand it. Well, the basic concepts of it, anyway. Even Einstein didn't fully understand -- just worked out the math and said, "Well, that's weird..."
We tend to see time as constant. Something that cannot be altered or changed. We see the sun and moon rise and set -- we see the seasons come and go. We even watch the stars and planets wheel around the sky. All those primitive forms of keeping time indicate, and logically so, that time is an "arrow." Always moving forward, in the same direction, at the same speed. And for everyday applications, that's a perfectly sound assumption.
The problem is that, just based on our tiny slice of universal observation, we don't have all the points of reference to know that time is always an arrow. Classical Newtonian physics works great for a falling apple, or the motion of a car. But when you start getting to something like the speed of light, or sizes that are sub-atomic, the laws of classical physics start turning into mashed potatoes.
For the sake of this particular post, I'll stick with velocity and how it relates to time. Basically, time is very malleable. This fluid nature of time isn't very apparent at relative speeds we experience day-to-day. If you are in a car, time is different for you than if you're walking. Though the difference is SO minute you'd never be able to measure it. Even flying in a 747, which goes, what, somewhere around 800 MPH? You can't tell the difference.
Even a space shuttle traveling at escape velocity, which is about 12 miles per second, doesn't make the difference very apparent. But over a relatively short period of time, that difference can be measured with instrumentation. And the faster you go, the more noticable it is -- approaching the speed of light. It's shaped like an exponential curve, where the X-axis is your speed, and the Y-axis is time dilation. At the speed of light -- C -- there is an asymptote. That, mathematically speaking, means that the speed of light is NOT attainable -- not by anything with any mass anyway. There's a mathematical formula, called the http://en.wikipedia.org/wiki/Lorentz_transformation which allows us to calculate the relative passage of time for someone traveling at high speeds compared to a "stationary" person (though in the universe, nothing is stationary).
The way it works is that all space and time are interrelated and connected in such a way that any change or impact made to one affects everything. In a specific sense, the faster you go, the more massive you become. Literally, the more physical mass you take on, which means more energy is needed to increase your speed another step. In that graph, the speed of light isn't attainable NOT because it's just too fast -- it's because something with mass traveling the speed of light would have an infinite amount of mass, and require an infinite amount of energy to reach that velocity. That's why only photons (or wave-particle energy) move at that speed.
So the faster you go, the heavier you get. The heavier you get, the more energy it takes to move you. Also, the heavier you get, the more spacetime itself warps around you (which is exactly what gravity is -- the more massive something is, the more "warping" of spacetime it does, which is, basically, what gravity is). The more spacetime warps, the more -- well -- spacetime warps. You can't technically separate the two because, as I said, all space and time is interconnected. Essentially, the faster you move, the greater the spacetime warping you have (or rather, the more powerful gravity well is created around you) because of the accrued mass.
That's how it works. Why it works is just a mystery of the universe. I guess it's just a law of nature. It's like asking why the universe is here. There is no "why," it just "is."