How did Greek drama originate?

Greek Drama grew out of religious celebrations (as much theatre has over the course of time).  Originally, there was a simple dithyramb, what we might call a hymn.  Then, in order to make these more interesting or relevant to the celebrants, the idea of acting it out came forward.  Thespis is credited with stepping out of the Chorus to do this acting (which is why we call actors Thespians), but there isn't really proof of this.  It's a good story, though!

Anyway - part of the way that the Gods were celebrated was in the retelling of their myths, and this became the subject matter of many of the Greek tragedies.  It began with the single actor, then Aeschylus added the second actor, and finally Sophocles added the third.  The Greek tragedians followed the three actor rule - no more than three actors on stage at any given time.  Euripedes is the third major Greek tragic playwright, and he is considered the most modern of the three, mainly because of his portrayal of women and skeptical treatment of the Gods. 

That's the nutshell version.  Greek Comedy also evolved out of those religious festivals, but dealt more in topical issues of the day (Old Greek Comedy) or domestic issues of the home (New Greek Comedy).