The Founding Fathers of the United States, and the Enlightenment philosophers they learned from - again, the people whose machine we are supposed to keep running - were obsessed with Greece and Rome. We have to understand where they were coming from.
In order to understand how our republic works, we need to understand the thoughts of the people who built it. If you're not a mechanic, you wouldn't try to fix your car without first trying to read some sort of instructions. This democracy we live in is like a piece of foreign machinery we are supposed to operate. Its governing institutions were imagined and bequeathed to us by a number of men, and all those men studied the history of Greece and Rome, as did the philosophers and writers and statesmen they took inspiration from, and those that these men took inspiration from. It's that we live in a time built by dead men who preceded us.
It's not just that history holds important lessons. There's a reason for that, and there's a reason why it's a shame we no longer do so. From the early Middle Ages until just a few decades ago, every educated person had to study the history of Greece and Rome.