Jesus seems to be always pushing the envelope of righteous behavior. Torah's ten commandments were honed - fulfilled, by Jesus' words - into a set of standards held by no other religious practice. "You heard it said, 'Do not murder,' but I say, don't even call a person a fool..." Egad! There are so many fools!
In the Sermon on the Mount, Jesus refines and fulfills the truest meaning of many commands, including the command not to take the Lord's name in vain.
You have heard it said to those of old, "You must not break your vows; you must carry out the vows you make to the Lord." But I say, do not make any vows! Do not say, "By heaven" because heaven is God's throne. And do not say, "By earth" because the earth is is his footstool... Do not even say, "By my head!" for you cannot turn one hair white or black. Instead, let your YES be YES and your NO be NO. Anything beyond this is from the evil one.
Here, the vow (or swear in KJV) means "as my witness." There cannot be any outside validation of the meaning of our words. I can be my only witness. I have to be good for my word; I must be true to my own words. My actions must follow. Today we would say, "Walk the talk."
The original commandment means using God's name in such a way that it is emptied of its value by assigning it to our behavior. Our behavior is our own. We are the only ones who can prove ourselves. Jesus wants us to be a standard bearer of our own actions, to take responsibility for what we say and do.
The use of hyperbole of needing to back our words or actions with an external source only proves two things. The first is that we are not good for our word. The second is that any attempt to do so is not of the godly order - it is from the evil one.
This fulfillment by Jesus is to push our definition of integrity, of righteousness. We are to be true to our nature and honest about it. We might call that transparency today. I say what I am and I am what I say.