It is in their tradition and it is something they strongly believe in.
At their temples.
Gods
At temples or by prayer.
Yes,some do!
Most did, yes.
To worship their various gods.
burnt offerings and temples of worship
They worshiped a whole pantheon of Gods.
Temples dedicated to specific gods or goddesses.
They burnt offerings and prayed and sacrificed animals.
Ancient Greeks needed a place to worship the gods and to do other things.
The ancient Greeks worshiped their gods for the same reason that any spiritual-religious people engage in worship: they believed that their gods were real, that those gods desired worship and reverence, and that those gods would reward those who worshiped them. While the Greek notion of "god" and "gods" changed significantly over the centuries, the fundamental reason for worshipping them did not.