In my arguments with people who do not believe that Time Dilation works, I've discovered that they do not seem to be able to distinguish between reality and Relativity.
Reality is about what is actually happening. Relativity is about what is perceived to be happening.
Einstein's 1905 paper "On the Electrodynamics of Moving Bodies" was intended to show that reality can be very different from what is perceived. The purpose of the paper was to explain two things: (1) The laws of physics are the same throughout the known universe, and (2) the speed of light is fixed and does not change if someone is moving. He was disproving a theory that the laws of physics can change under certain circumstances and the theory that the speed of light can change under certain circumstances. He explained that it is only one's perception that changes under those circumstances.
Time Dilation works in reality. If a clock, person or object moves very fast (or moves closer to a massive gravity source), time will slow down for that clock, person or object.
In the famous (or infamous) "Twin Paradox" hypothesis created by Paul Langevin (but usually blamed on Albert Einstein), one twin travels in a space ship at speeds near the speed of light while the other twin remains stationary on Earth. When the traveling twin returns to earth, he is much younger than the stationary twin. That hypothesis was based upon Albert Einstein's clock hypothesis in his 1905 paper which says,
If at the points A and B of K there are stationary clocks which, viewed in the stationary system, are synchronous; and if the clock at A is moved with the velocity v along the line AB to B, then on its arrival at B the two clocks no longer synchronize, but the clock moved from A to B lags behind the other which has remained at BIn other words, you start with two stationary clocks (A & B) that show the exact same time.
Clock A is moved at high velocity to be beside Clock B.
Clock A no longer shows the same time as Clock B. Clock A shows less time has passed.
How much less time has passed depends upon the velocity of the move.
This experiment works in reality and has been repeatedly demonstrated with atomic clocks of various kinds. Click HERE, HERE and HERE for examples.
Relativity is about how effects like Time Dilation are perceived while moving very fast. The twin who took the trip at speeds near the speed of light would have perceived that everything aboard the space ship was working normally at normal speeds. If he has no windows on his space ship, he would not perceive any of the effects of the slowing down of time. That does not mean the traveling twin will believe that he is standing still or that time is passing at the same rate for him as for his brother. He'll know what is actually happening, just as, in today's world, if he was in an airplane flying at 550 miles per hour, time for him would be running slower a fraction of a microsecond per hour than for someone on the ground. These examples are just illustrations to show that someone can be moving very fast on a space ship while the visible evidence may indicate that he is not moving at all. And time may have slowed down for him, but he'll perceive no effects of it while knowing that the effects will be very apparent when he returns to earth.
The people with whom I've been arguing seem to become confused when a scientist explains how things are perceived when one person is moving and the other is stationary, or when two people view each other while both are moving. The problem seems to be that scientists make no attempt to explain what is actually happening when discussing Relativity. They only talk about what is perceived to be happening. And they let those listeners who are confused fall by the wayside.
The biggest arguments are over an imaginary event where two space ships pass each other in some remote and empty part of space. Each pilot perceives his ship as stationary and the other ship as moving. Each pilot perceives the clocks aboard his ship as working normally, while the other ship's clocks may be perceived as running slow. In reality, of course, each pilot would know his speed relative to earth (and to the speed of light) and he'll be able to calculate the other pilot's speed.
Each pilot will also know that, even though all the clocks on his ship are perceived to be operating normally, they are all running slow relative to clocks on earth.
And he'll know the pilot on the other space ship is perceiving similar things aboard his space ship.
At least, that's how I understand Time Dilation and Special Relativity.