The holiday commemorates the victory in the fight for independence from the Syriac branch of the Alexander's crumbling Greek empire. Though the Jews had great respect for the Greek philosophers and honored Alexander, those who came after him tried to impose incompatible customs and Greek paganism.
When the Temple in Jerusalem was purified and re-dedicated (after removing the Greek idols and cleaning the alter from the pigs the Hellenists sacrificed), the legend is that only one day's worth of holy oil was found -- but it miraculously burned for eight days until more could be produced. To commemorate this miracle, we eat oily foods (e.g. potato pancakes and doughnuts).
However, I tell people that we eat oily food to commemorate our victory over Greece (grease).