“Respect my authoritah!”
~ E. Cartman
First, Exousia has several meanings. Likely in the days of the King James translation the words most popular translation meant “Authority over mankind”.
In Hebrews, Paul tells us that Christ is The Great High Priest. This is a significant title. It was the High Priest in the Temple at the time of Christ who authorized sacrifice. If you didn't hold the priesthood you were not authorized to offer sacrifices. Today in the Roman Catholic Church and the churches derived from it, only the priesthood can prepare and offer the sacraments to the people. There are cases where authority is important, the authority that is important comes from God.
I hope it doesn't sound like I'm getting away from the subject here but the point is that Christ was The Great High Priest and offered Himself a sacrifice for all the people.
The authority we are speaking of that the people recognized at the Sermon on the Mount was that. Christ spoke as one with authority over mankind.
There were many times that Christ preached and referred to scripture for example but here He said “I say”. The Jewish Rabbi would never say I say because he had no authority to do so, he would always refer to scripture or tradition but never his own words. Christ on the other hand was the author of The Scriptures, the Great Law Giver and the only name by which mankind can be saved. He has authority over us.
I hope I haven't offended anyone, if I have I beg your pardon.