Illustration of a collision between two planetary bodies

How the moon came into existence, there are many theories behind that. But the most approved theory among the scientists is Giant-impact hypothesis, sometimes called Big Splash.

The Giant-impact hypothesis tells that the Moon came into existence with the impact between the Earth and a celestial body (generally called Theia) approximately 4.5 billion years ago. 

That impact made a small piece of Earth fly away into the sky and that small part is nothing but  the Moon.

Theia comes from Greek mythology, under the name of Titan, who was the mother of Selen, goddess of the Moon.