Why do most of the myths behind constellations originate from Greek? Why do most of the myths regarding constellations happen to have begun from Greek civilization? Why not other civilizations? 

Most people who lived in the past era probably observed star patterns in the sky and created their own constellation myths. 



Also, I would like to know more about some of the stories regarding the myths of those constellations.
Ptolemy was Greek.  He wrote an important book called the Almagest. In it, he lists and describes the constellations as used in Greece in the 4th century BC. Naturally, the Greeks used elements from Greek Mythology in their sky culture, just as Chinese sky culture uses elements from Chinese legend and myth.

Ptolemy's Almagest and Greek sky culture influenced the Hellenistic world, and through Rome, it influenced the whole of Western Europe.  Then through European expansion, it came to influence the whole world.

There are several additions to Ptolemy's system: The Southern Sky was divided into constellations by Europeans in the 15th century, and there are several additions to the Northern sky, such as Coma Berencies (Hair of Berenice II of Egypt) and Camelopardalis, the giraffe, (added in the 15th century)