Jesus is referring to the ceremonious way in which Jews, particularly the Pharisaic types, took oaths. It was one of the things in the culture of the time. Jesus was nailing it as an undesirable procedure because it was all about pulling the wool over other people's eyes. That is, people swore these impressive sounding oaths, and the whole situation they were creating thereby was phony.
What were the oaths anyway? They were forms of words designed to impress. They weren't indications that the person swearing the oath had any power over the things that the oath mentioned. It was just using swear words to impress and so produce acceptance of things in relation to the oath, when in fact this was relationally phony because the persons taking the oath to impress didn't intend to keep the promises or commitments that they were apparently confirming by taking some oath. They were just fooling folks. And Jesus said, in essence, "Don't let this foolery ever touch you."
This fits really well into the context of the Sermon on the Mount because it's going after the Pharisees for their hypocrisies. Jesus is against hypocrisy in all its forms, and he's for straightforwardness and transparency and honesty and responsibility in all its forms.
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