The essence of time dilation with respect to the theory of relativity has been demonstrated experimentally. In one experiment, two atomic clocks were carried on high speed air plains. One traveled eastward (with the rotation of earth), and one westward (against the rotation of earth). After the flight, these clocks were compared and found to have either gained or lost timed (relative to the control clock which remained on the ground). The amount of time lost and gained was a function of their direction of travel (motion), and altitude (gravity). The results confirmed the predictions of time dilation from Einstein's theory of relativity.
"During October, 1971, four cesium atomic beam clocks were flown on regularly scheduled commercial jet flights around the world twice, once eastward and once westward, to test Einstein's theory of relativity with macroscopic clocks. From the actual flight paths of each trip, the theory predicted that the flying clocks, compared with reference clocks at the U.S. Naval Observatory, should have lost 40+/-23 nanoseconds during the eastward trip and should have gained 275+/-21 nanoseconds during the westward trip ... Relative to the atomic time scale of the U.S. Naval Observatory, the flying clocks lost 59+/-10 nanoseconds during the eastward trip and gained 273+/-7 nanosecond during the westward trip, where the errors are the corresponding standard deviations. These results provide an unambiguous empirical resolution of the famous clock "paradox" with macroscopic clocks."

J.C. Hafele and R. E. Keating, Science 177, 166 (1972)

The Twin "Paradox" (not really a paradox) explains this idea also, and more completely in the three planets thought experiment. The key point being inertial reference frames.

Common sense likens photons to bullets that are shot out of a gun. A bullet shot out of a car in potion travels faster than one shot from a stationary refrence point. The difficultly with this analogy is that this does not apply to things moving at the speed of light (bullets move much slower than photons - which always travel at the speed of light for whatever medium they are in). The speed of light is constant. Period. Yes, it does get confusing with the concept of inertial reference frames.

Let us imagine a clock in which time is measured by a photon bouncing between two surfaces:

As your velocity approaches c, this clock will start to slow down. Why? Simple, because the distance between the two surfaces grows - they are no longer directly on top of each other:
-----    -----
  \        /\
   \      /  \
    o    o    o
     \  /      \
      \/        \etc...
However, you will not notice the clock slowing down because you are slowing down with it. To actualy see the slow down (or speed up) you will need to be in a diffrent frame of refrence.

Time dilation will happen in accordance to the Lorentz transformation. Yes, this is a big 'nasty' function with lots of primes and radicals and subscripts.

c is the speed-limit of light in that nothing can be accelerated to the speed of light with respect to you. Accelerating two objects to 75% of the speed of light in different directions giving a relative velocity greater than c just doesn't work (see the three planets thought experiment for more information on this). Space-time is interwoven in such a way that no observer ever sees an object moving toward them at greater than c. Once the Einstein velocity addition is applied, the result is always a relative velocity less than c. This may not agree with common sense, but this appears to be the way the universe works.