What is the ruling on a wife refusing to go to her family, and what should she do, after her husband swore to her saying: "Divorce be upon me if you do not go to your family tomorrow, then you are forbidden to me"?

The ruling on the oath your husband swore depends on his intention when he made the prohibition. If he intended divorce, then it is a divorce. If he intended ẓihār (a form of divorce by comparing one's wife to a forbidden female relative), then it is ẓihār. If he intended an oath, or did not intend anything specific, then it is an oath, like swearing by Allah Almighty. If he intended divorce, it takes effect if you go to your family. If he intended ẓihār, it takes effect, and it becomes forbidden for him to have intercourse with you until he performs the expiation for ẓihār. If he did not intend divorce or ẓihār, then he must offer expiation for an oath. However, if you go to your family's house, then there is nothing binding on him. Your husband must consult scholars to ascertain his intention and act according to their fatwa, knowing that swearing by divorce and prohibition is impermissible.