When Jesus says, "You have learnt how it was said," he is making a clear reference to the Old Covenant, the Law of Moses.
That Law gave the Jewish people their unique standing among all the nations of the world, because God himself had given it to them - God's finger had inscribed the tablets of the law.
For 1,500 years, Israel's prophets and rabbis had interpreted it, applied it to changing circumstances, and exhorted the people to live it out.
But never in those fifteen centuries had a faithful Israelite ever claimed authority over it. After all, the Law had come directly from the Lord, so who could possibly have authority over it?
So, when Jesus says, "… but I say to you…" implying an addition to the Law, his listeners are faced with something entirely new, someone who claims authority over the Law of Moses.
Jesus is requiring of them a new allegiance and making way for a New Covenant. The Sermon on the Mount was revolutionary not only in its ideas, but in the claims of the Lord who gave it.
And this claim, this implicit claim to have authority over divine law and therefore to actually be divine, has consequences. It means that his commands demand obedience.
In the ancient world, obedience to a ruler was a familiar concept. In today's world, dominated by political democracies, it has become less so. In fact, today's critical, self-sufficient, democratic mindset (so useful for politics) can even seep into the Church (where it's much less useful).
But the truth of Christ doesn't change with fashions and referendums. In our relationship with Jesus and his Church, humble obedience to legitimate authority is a virtue, not a vice.