Time dilation is just a property of time as Einstein related time, force, gravity, speed, and distance in both Special Relativity and General Relativity (which includes gravity). Time stretches in relation to speed and gravitational fields. The formula to calculate it are straightforward just like the ones to calculate speed and distance.

The way a GPS works is to send a clock and a radio into orbit and connect it to a clock and a radio on Earth. The clock measures the time it takes the radio signal to reach the other radio. Since the clock is in orbit at a high speed and it has less gravity (being far from the Earth) time operates at a different rate. The GPS calculates the change in time itself to adjust the time it takes the signal to travel in order to give the correct answer and therefore locate your GPS accurately. Without the time dilation factor, your GPS wouldn't work.