Does the husband’s divorce take effect if he swears to his wife by divorce that she will not enter her family's home, after he saw her getting into a car with her cousin’s son?

Praise be to Allah for guiding you to regret your mistake, for regret is repentance. Your husband's anger is justified and he will be rewarded for it, as it indicates his praiseworthy jealousy. As for your family, their blame is justified for their mistake in escalating matters and speaking about Allah without knowledge, which is among the major sins.

As for your husband swearing by divorce that you will not enter your family's house, there are two mistakes in this: swearing by other than Allah, and preventing you from your kin. The ruling on his oath is as follows: if he intended to effect divorce, it occurs upon your entry. If he intended a deterrent, the majority of scholars also believe that divorce occurs. However, Sheikh Al-Islam Ibn Taymiyyah and his student Ibn Al-Qayyim hold that it does not occur and he must pay an expiation for breaking an oath. The preponderant opinion is that divorce occurs when the sworn-upon condition is met, regardless of the swearer's intention.