This never made any sense to me whatsoever.
I’ve see all the physicists (Michio Kaku, Stephen Hawking, Neil DeGrasse Tyson, etc.) explain this principle but it doesn’t make sense. They say that if you were to go to the moon and back at a certain speed near the speed of light, you might return to Earth a thousand years into the future like what happened in Planet of the Apes. But if you were going at the speed of light, you would arrive at the time light takes to arrive there. Why the dip? What is being missed?
So, people seem to have explained how the time dilation works.
Now to the why:
We found out that the speed of light in a vacuum never changes. It is always the same. Usually in our everyday life speeds add up.
Let’s say you have a cannon that shoots out a ball at 100 km/h. It would hit a wall at 100 km/h.
If you were now driving a car at 100 km/h and shot the ball out of that it would hit the wall at 200 km/h.
But the speed of light is different. If you had a light cannon to shoot some light at the wall it would hit at 1 c (the speed of light). But if you drove a car at 100 km/h and shot the cannon from there it would not hit the wall at 1c + 100 km/h. It would still be 1 c.
That’s pretty strange and counter intuitive. Albert Einstein calculated the results of that. What must really be happening is that time slows down and space contracts slightly the faster you drive. So that in the end the speed of light stays the same.
We usually don’t notice that effect at the speeds of your everday life. But we have satellites for GPS with atomic clocks in orbit where that effect becomes relevant. By now they are a few seconds out of sync with stationary earth clocks.
So GPS navigation proofs that time dilation is real.