Time dilation is a result of a fundamental symmetry of the universe.

Start with good old Newtonian motion. Suppose I see some object move a distance $(dx, dy, dz)$, then [Pythogoras' theorem][1] tells me that the total distance it has moved, $ds$, is given by:

$$ ds^2 = dx^2 + dy^2 + dz^2 \tag{1} $$

Now suppose you're using a coordinates rotated relative to mine (maybe we're in cities at different latitudes). The individual values for $dx$, $dy$ and $dz$ that you measure won't be the same as mine. But if you use equation (1) to calculate $ds$ you'll get the same answer as I do. So we have a symmetry that the distance $ds$ is the same for all observers. We say $ds$ is an **invariant**.

Actually, this should be obvious. The length $ds$ could just be the length of a rod, and the length of  rod doesn't change if you rotate it or move it around (well, not in Newtonian physics ...).

What special relativity says is that equation (1) needs to be extended to include movements in time as well, and the new form is:

$$ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 \tag{2} $$

(we multiply the time $dt$ by the velocity $c$ to turn it into a length in light seconds $cdt$ - that way we are adding lengths together). And again special relativity says $ds$ is an invariant and all observers will agree on its value. But note a key difference between equations (1) and (2) - in equation (2) we have a minus sign in front of $c^2dt^2$.

Equation (2) describes a fundamental symmetry of the universe called [Lorentz covariance] [2] and it's this symmetry that requires time dilation to happen. Why Lorentz covariance should be a fundamental symmetry of the universe I don't know. That's just the way the universe is. To see how time dilation (and length contraction) arise from Lorentz covariance have a look at the question https://physics.stackexchange.com/questions/80511/how-do-i-derive-the-lorentz-contraction-from-the-invariant-interval.


  [1]: http://en.wikipedia.org/wiki/Pythagorean_theorem
  [2]: http://en.wikipedia.org/wiki/Lorentz_covariance