Greece is considered by many historians to be the birthplace of Western Civilization because many aspects of our modern societies originated in Greece. Let us look at a few of these aspects.
One thing that began in Greece was Western art. The Greeks engaged in the creation of art of all kinds. They had painting and sculpture. They had poetry. They are the first people we know of to write plays. Their arts influence our arts even to the present.
Some of our intellectual tradition begins in Greece as well. The two main intellectual fields that go back to the Greeks are philosophy and history. Greek philosophers like Socrates and Plato are among the most important philosophers in the history of the world. Herodotus is generally seen as the first real historian known to us. Some aspects of the sciences also got their beginning in Greece.
Perhaps most importantly, though, the Greeks were the originators of democracy. Athens is particularly famous for its democracy. Because democracy has come to dominate the Western world today, Greece is particularly important.
For all of these reasons and more, Greece is often seen as the birthplace of Western Civilization.