How did ancient Greece “set the stage” for contemporary theatre?

Modern theater, which can trace its roots to Greece and the dithyrambs performed in honor of the god, Dionysus, owes much of its vocabulary, architecture, and even types of work to the ancient Greeks.
According to Aristotle in his work, Poetics, which is the earliest extant account of the origins of theater in Greece, theatrical performances began to take shape in Athens circa 600 B.C. Performances were held in an open-air amphitheater. While many modern versions of this are covered, many venues employ this arena-type seating model. The stage or performance area was called a Proscenio. Today, the word, “proscenium,” refers to a particular type of stage and still resembles its Greek counterpart. One of the first actors was named Thespis, and it is from him that the word, “thespian,” which is now a synonym for actor, originated.
Many trappings of Greek theater are still in use today: masks, painted sets, and costumes, as are modern versions of ancient special effects devices. For instance, the Aeorema or crane-like device that created the illusion of flying gods in ancient Greece bears some resemblance to modern theatrical rigging.
The Greeks invented the three main types of plays that are still in use today: comedy, tragedy, and satire. They also expanded the concept of acting by developing dramatic action, conflict, and creating more naturalistic forms of expression.
http://www.classics.upenn.edu/myth/php/tragedy/index.php?page=theater

https://www.reed.edu/humanities/110Tech/Theater.html