You see dragons appearing in art from Latin American, East Asian, European, and African cultures, but how did they all develop this legend?
There's always room for discussion, but perhaps the section Here be dragons will answer your inquiry.
This kind of overlaps with the Dinosaur question....
In addition to the Dinosaur theory, I will mention another theory here. There are certain magical and mythological themes that recur widely among ancient cultures spread across the globe. This subject is covered in much detail in "The Golden Bough" by Sir James George Frazer. The idea is that 'primitive' (IE non-rational/scientific) cultures engage in 'magical' thinking, which is replace later by scientific thought. What is appropriate for this thread is the strange way in which the magical beliefs, embodied by myths, are often eerily similar.
This is similar (in a way) to what C.G. Jung called 'archetypes' of the 'collective unconscious'. While I don't exactly believe in Jung's own conclusions on this, I do find fascinating and revealing the fact that Jung identified, similar to Frazer, that humans often share symbols across cultures, time, and space.
What causes this? Science of the mind is still in a fairly primitive state, although new tools like MRI's are for the first time providing a window into how the brains structures function. My guess is that these archetypal images reflect a structural process embedded in the human brain itself - possibly as a result of the evolutionary process. (Why are humans afraid of spiders when we are so much bigger than they are? Is it because we evolved from much smaller mammals?)