Imagine I have a measuring stick. The length is irrelevant but suppose it's long enough so the phenomenon mentioned here is visible. I take this stick and measure out a specific distance.
Due to the material that stick is made from, forces such as variances in heat cause my stick to alter shape slightly, becoming longer or shorter. With this altered device, I go back to measure the same location.
Lo and behold, I find that same location I previously measured has shrunk or grown ever so slightly.
Well, actually, the location didn't change (at least for purposes of this example) but the device I used to measure it did. This is why we don't make yard sticks out of rubber bands.
Now we turn our attention to atomic clocks. These devices use the movement of electrons to measure the passage of time. (the actual technique is much more complicated, but you get the idea.) These are highly precise, and we have found that slight variations of forces such as gravity, etc. can actually alter time.
Well, perhaps that's not really true. Those forces altered our measuring devices, and we mistake that for meaning they altered what was measured. In this case, time. Time was not altered. Simply the measuring stick was.
It is a subtle difference, but one that is quite important if we are to derive the correct hypothetical information from them.