When I was a kid and would read about eclipses, all the talk of umbras and penumbras made my eyes glaze over a little bit, but it's a concept you really need to have if you want to understand why this works.
When something has a blurry shadow, the blurry part is called the penumbra, and it's there because the light source is larger than a point. If you're in the penumbra, you can see part of the light source peeking out from behind the shadow caster. The fully dark part is the umbra. In the umbra, you can't see the light source at all.
If you think about it, the larger the light source, the smaller the umbra (the dark part) is. That's because some of the light from the part of the light source that sticks out the farthest angles back inward and illuminates area that would otherwise be in full shadow (i.e., in the umbra). If you shrink the light source, the umbra gets bigger.
So think about what happens when an object close to the light source moves so that its shadow overlaps the shadow of a farther object. From the point of view of the surface on which the shadow is cast, the moving object (the one closer to the light source) is effectively making the light source smaller. So the umbra (again, that's the dark part) of the object farther from the light source gets bigger just where the moving object's shadow is.
I should say that I learned this explanation just now by watching the animation in the linked Wikipedia article, but I thought it would help to put it into words.