Gods and goddesses were important because they embodied certain characteristics, qualities and sometimes imperfections that served as inspiration and examples for how individuals should live their lives and for how society should organize itself.
Especially when we are talking about ancient peoples, myths explain natural phenomena and life cycles, so frequently there were certain gods and goddesses whose characteristics and habits were directly linked to where given society lived. For example, the Mesopotamians explained the constant floods of their territory's rivers as coming from the god Enlil's anger.
Gods and goddesses are different between different belief systems but it’s possible to perceive some similarities: immortality, great strength, the personification of natural forces, realization of great deeds.
Mythological beliefs shaped and were shaped by culture. These beliefs created a series of rituals throughout the lives of the societies determining when people were to be married, to go to war, to harvest, etc., but they were also shaped by where these people lived, the kind of economic activities they made and so on.
The articulation between structures and mythological beliefs is something we need to consider each case individually, although most of the historical periods we know show us that mythological beliefs, despite their great importance, aren’t enough to shape alone an entire society. Structures result from an interaction between economic, political and social forces, where beliefs are but one of them.
Myths were important for these civilizations because they explained how the world came to be, how it was created, and why it worked that way. They were also important as an explanation for the position those civilizations occupied in the universe and in history. As for why these religious places were public, when we talk about Greek temples we should understand that Greek religion was a civic religion: religious rituals were used to gather people and to create sense of belonging to something greater than themselves. Usually religious places were built to delimitate territory, to please a god or goddess, or to show some leader's strength like the Pharaohs.
Both Aztecs and Greeks honored their gods by trying to be as great as them. When someone showed great physical strength, beauty, etc., that was an homage to their gods.
These stories should demonstrate, for example, the greatness of a god, or how periculous it was to follow some kind of choice. Violence was present in these stories to make them worthy of gods and goddesses and in order to work as examples to be followed (or not followed).
Since the final years of the XVIII century we believe the study of history to be fundamental in order to understand how the world changes so we can make choices consciously. For example: once we understand how politics works by observing past coups and revolutions we can be better prepared to live moments where these things may happen again. So knowing the stories ancient peoples believed helps us understand how they thought and acted and helps us decide how to act today.
Myths belong to the sacred dimension of life, something found in every human culture. It is of the nature of the sacred to possess fantastic elements since it explains life and the world in a different manner from rationalism, for example.