Why does a shadow appear blurred in distance?

As another answer pointed out, a shadow is a place where (approximately) no light reaches. This is why a shadow is black; it's colorless(light is a mixture of colors). read more

Sure, there is some very small effect of this, but the two reasons mentioned above is the main reasons that does a huge impact on the anatomy of the edges of the shadow. The latin word for the edge is by the way called “penumbra”, and the penumbra can be blurred in different directions due to different width of the light source. read more

So as distance from a spherically emitting light source doubles, the shadow’s area quadruples. The quadrupling comes from a doubling of the shadow in the x and y directions, so if an upright stick with a shadow of 2 feet long is moved twice as far from the light source, the stick’s shadow is 4 feet long. read more

It's size is about 1/100th of its distance to Earth, so the soft part of the shadow (penumbra) will have depth of 1/100th of the distance the object is from the background. E.g. a ball 1m above ground will cast a shadow with 1cm penumbra if Sun is directly above. read more