Why Did They Enter a Curse?

This is a reference to the law of Moses.

See, I set before you this day a blessing and a curse: a blessing for obeying the commandments of the LORD, your God, which I give you today; a curse if you do not obey the commandments of the LORD, your God, but turn aside from the way I command you today, to go after other gods, whom you do not know (Deut. 11:26-28).

I call heaven and earth today to witness against you: I have set before you life and death, the blessing and the curse. Choose life, then, that you and your descendants may live (Deut. 30:19).

To enter into the covenant meant blessings for those who kept it, but bad things would befall those who broke it.

In Nehemiah, what is being described is the Jewish people in exile. The Jewish people had broken the covenant and had been conquered and dispersed. Some of the Jewish people in exile in the Persian empire decided to solemnly recommit themselves to the covenant. In this solemn ceremony, they were recommitting that they understand the covenant contains blessings but also responsibilities for them to fulfill.