Leia's identity is part of a lifelong secret, and she intuitively understands this
Luke (and the audience) know her identity is a great secret for her protection, as it is explained in the scene on Dagobah when Luke converses with the ghost of Kenobi.
BEN: Hmm. To protect you both from the Emperor, you were hidden from your father when you were born. The Emperor knew, as I did, if Anakin were to have any offspring, they would be a threat to him. That is the reason why your sister remains safely anonymous.
...
LUKE: But you can't let her get involved now, Ben. Vader will destroy her.
So Luke definitely knows to keep their relationship quiet. The question is-- how does Leia know? All he did was tell her that she was his brother;sister; he did not tell her to keep it secret or why it would be necessary. So it is a good question.
The cynical explanation is that it is bad writing, and the screenwriter failed to differentiate between the audience's knowledge and Leia's.
A more charitable explanation can be discerned from Leia's immediate reaction to the astounding news:
LEIA: I know. Somehow...I've always known.
She immediately understands that she and Luke are involved in an epic, lifelong secret-- a secret that would be between only the two of them. And she keeps that secret steadfastly, until the end of the film when the Emperor is dead and the danger is gone.