The reason I don't like the usual time dilation explanations is that they make it too easy to confuse coordinate time with proper time. Coordinate time is the time coordinate, that goes along with the space coordinates, to make up the spacetime coordinates for an event (x,y,z,t). The coordinate time depends on the observer, as do all of the spacetime coordinates for any event. So, for any given event, there are as many different coordinate times associated with it as there are observers.

But there is only one proper time for a given event, and that is the time ticked off by a clock in the rest frame of the event. That time, as observed by an observer at rest with respect to the clock, never dilates, whatever anyone else might see or think.

We see the muons falling through the atmosphere, and we see "time dilation", the muon lives longer in our reference frame. But the muon does not see it that way. It's too easy to think of time dilation as "real", not in the sense that we see it, but that it somehow means more than what any other observer would see. I call it an illusion, because it is entirely dependent on the observer reference frame, which is not the case for proper time.