There are many ways to explain it, but here's one I remember as being more tangible:
Say you're driving along a country road at 60mph. You come to a sign saying that the speed limit is 40mph, so you "remove" 20mph from you speed (i.e. 60 - 20 = 40).
You come to end of the 40mph zone and the speed limit goes back to the national speed limit, single carriageway, i.e. 60 mph. You increase speed, thus "removing" the "-20" that you imposed earlier, i.e.
-20 - (-20) = -20 + 20 = 0.
Does that make sense?
If you want a more "mathsy" answer, consider the equation -5 -(-3):
-5 = -1 x 5 (i.e. -1 times 5)
-3 = -1 x 3
So we have
-5 - (-3) = (-1 x 5) - (-1 x 3)
Take the "-1" out of the equation:
(-1 x 5) - (-1 x 3) = -1 x (5 - 3) = -1 x (2) = -2
So -5 - (-3) = -2
It's not the easiest of concepts to grasp, but it is a fundamental one.