The Talmud made a hash of things with vows and promises, creating a whole category of vows that had no effect, and allowed the smart-aleck rabbis to get away with lying. The New Testament makes a point that you shouldn’t need a vow in the first place, that people should be able to trust your word on the matters great and small.
In this case, it has to do with making a vow of obligation to the Lord. In most cases that refers to including God as witness of some promise you make to anyone else. In Matthew 5:33-37, Jesus specifically refers to this chapter in Numbers. He said vows and oaths should be unnecessary, that people should stop cheapening the whole process of communication under the Covenant.
Still, in the Ancient Near Eastern culture, people would typically bind themselves under oaths for any number of reasons, but mostly because ANE folk were skeptical and cynical in the first place. At any rate, God tolerated a certain amount of this, but the basic requirement is that people of the Covenant follow through unless released from the promise by the beneficiary, or by some higher authority.
This is the nature of small “c” covenants between people in that culture. If you were wise enough to operate from the heart, you would not likely bind yourself in folly. However, this was a feudal society and higher authority would have to be made aware for most covenants to stand. This chapter assumes things that don’t interfere with a higher feudal authority. Thus, a man should bear the price of keeping his promises.
However, a woman was protected from her folly. The reason was stated clearly in the matter of the Fall: women can be morally deceived more easily than men. Thus, this passage warns that any vow a woman makes under patriarchal authority requires her covering’s acquiescence. Not his permission, you notice, but simply that he is aware and had nothing to say about it.
This holds for a woman whose covering is either her father or her husband. A widow or divorcee without patriarchal covering is stuck. Of course, in modern Western societies, the state stands in for her covering, but God doesn’t buy that. His law stands on Creation itself. While the consequences may not come right away, Creation itself marks the defilement of manipulation and deception.
I’ve noticed that about women, and effeminate men. They tend to go with the flow created by the strongest influential presence, good or bad. Usually that presence is male. This by itself isn’t an indication of a practical weakness, necessarily. There is a definite asset in having people support you more readily than not.