Why did the Ancient Greeks love tragic plays so much, rather than any other type of play?
We must start with two caveats. First, it is not clear that the ancient Greeks loved tragedy more than comedy. They had competitions for both and men like Aristophanes were no less prominent than men like Sophocles. Second, we cannot objectively know why the Greeks liked either of these genres. It would be hard enough to objectively know why Americans like reality shows and much harder still to understand the mindset of people from 2500 years ago.
That said, some scholars argue that Greek drama was so popular because of the way it addressed tensions in Greek life and Greek religious belief. For example, tragedies tended to examine the relationship between people and the gods. They would explore the way in which people reacted to the actions of the gods. This was, to them, an important part of everyday life. The plays also addressed issues that arose in the polis. The tensions that arose in everyday society would also have been important to the audiences. In this way, the tragedies were commentaries on issues that would have been important to the Greek populace. We can infer that they were popular because of their relevance to their audiences' lives.