As I have myself struggled a bit with this concept, I would like to present my own explanation of it. Context: Loss of significance is a loss of precision, not necessarily accuracy.
And for a long time, I didn't know the difference between the two concepts.
Explanation: Now first, I show them a calculation of subtracting two numbers close to each other in floating point. We then try to calculate the initial and final relative and absolute errors.
We then observe that the relative error has become much larger, and in addition to that, we have lost many significant figures.
Context: when I was presented this in my first course of numerical analysis, I said: "so? Who cares? We are still close to the number!","the absolute error is still bounded by the addition of the errors!"
But then when I realized the difference between precision and accuracy, I convinced myself that this was a loss of precision.
Explanation: the relativity of precision:
Intuitively, we can observe that as we increase the size of the Actual number to be estimated A, relative to the size of the absolute error, we see that the cluster of numbers (Whose average is a measure of precision) gets "closer together", and that might give us sufficient precision.
Example: Like trying to hit a ship with rockets, an error area of 1 meter is amazing.
But hitting an apple with a bow and arrow , and 1 meter is too imprecise, too big to hit the apple consistently.
Last part of explanation: I then say," we want to be consistent with our data, and a loss of relative error takes that away from us. So we "care" about loss of significance.
I myself am only partially satisfied with this explanation, and find that there is something missing, or perhaps it is in part incorrect.
I would appreciate your comments or any guidance from your part.