Updated: Jan 20, 2018
“Traveling close to the speed of light you would hardly age at all, but your friends and your relatives back home would be aging at the usual rate. When you returned from your relativistic journey, what a difference there would be between your friends and you, they having aged decades, say, and you having aged hardly at all! Traveling close to the speed of light is a kind of [magical potion] of life.”
- Carl Sagan, Cosmos, pg 218
What Carl Sagan is talking about here is Einstein's theory of special relativity. According to special relativity, when a clock moves through space it will run slower. So for example, if you are walking the busy streets of New York City and you walk past a person who is sitting on a bench, your watch will move slower than his watch because you are in motion and he is not. Why can't the person sitting on the bench notice that your watch is moving slower than his own (excluding the obvious reason that he doesn't have X-ray vision)?
The answer is because your walking pace is so slow that it has a very slow effect on time. We don't notice these things because the speeds that humans interact with each other in everyday life aren't fast enough. It only works at speeds that approach the speed of light, which is 670 million miles per hour.
If you hopped into a space ship and traveled at 50% of the speed of light while that person continued to remain sitting on the park bench, he could then notice the slowing of time on your watch. Your watch moving slower also means that you are aging slower as well, so once you returned from your journey through space at 50% the speed of light, he would want to know why you look so young compared to him.
This effect of time slowing down as we approach the speed of light is known as time dilation, and it has been tested by Matti Kaivola and his team at the Helsinki University of Technology in 1985, an experimental team at Colorado State University in 1992, and at the Max Planck Institute for Nuclear Physics in Heidelberg in 2005. The result? All of the experiments complied with Einstein's theory of special relativity.