Why this shape? Why do we see it when we measure something as simple and "random" as throwing dice, or as natural and complex as weather? Why, indeed. This is were the word "believe" comes in. The shape is persistent and useful to us, just like gravity, whether or not we understand its deeper roots.

(The source of natural variation indeed runs deep, as a principle of measurment.
For a innovative treatment of the source and role of "randomness", I recommend Stephen Wolfram's "A New Kind of Science")

Very simply, if a process is creating measurements that looks like a normal distribution, it is in "control" at the point where it is measured. (The middle and tails of the curve may not be where you'd like them to be, but at least you can predict that; or, as a teacher of mine once put it, "you're reliably manufacturing waste".)

• The roll of dice. 
• The size of maple leaves. 
• The thickness of Oreos (my favorite classroom exercise). 
• The caliper of trees. 
• The weight of candy bars. 

There is no substitute for direct experience in coming to believe in the normal distribution.  If you don't "feel" this one, measure and graph things until you do,