the main difficulty in explaining this phenomenon is because often people don't remember that time dilation is directional, so all they remember is silly apparent padadoxes like one of the two twins decides to take a flight at near speed of light and comes back in 20 years, which one will look older? rather than sticking to basic geometry of how light moves as seen by each observer.Explaining time dilation to disinterested students years later only added to the mystery.
(I should add trying to explain, it's inexplicable)
GPS relativistic time correction is a good example to get people to understand time dilation.