under what "curse" does the narrator claim to have been born? why is this significant?

At his core, it seems like Equality 7-2521 has always been a scientist. Since he was a kid, he's wanted to know about the world and its mysteries, and about life and its meaning. Not even the strongest discouragements of his society can quiet that desire, which he calls his "curse":

And questions give us no rest. We know not why our curse makes us seek we know not what, ever and ever. But we cannot resist it. It whispers to us that there are great things on this earth of ours, and that we can know them if we try, and that we must know them. We ask, why must we know, but it has no answer to give us. We must know that we may know. (1.25)

Equality 7-2521 was born with a curse: he's been thinking forbidden thoughts for most of his life. And he doesn't resist them.

This is bad, because according to the World Council, all men must strive to be completely alike. All people must form one big, happy, indivisible "WE."