The following is a snip from the book Olympiads: A Mathematical Olympiad Approach:
The obvious approach (for me) is to use the triangle inequality to get $$|a|-|b| \leq |a-b|$$ $$|b|-|a| \leq |b-a| = |a-b|$$
And the desired can be achieved by combining the two.
However, the given solution seems to do this much more succinctly. I have looked at this cryptic solution several times, but each time I've come back to it, I've never been able to make sense of it. Any help?