I don't get it. The early portions of the game says Baba Yaga cursed the baron's children, but as you make progress, you learn things that's seem to contradict that.

You learn that it was not Baba Yaga who cursed the son, but the Kobold. And the Kobold wasn't even doing it for no reason. The baron's son walked into his cave and asked the Kobold to bow down to him because he was the baron's son. The Kobold wasn't having any of that (nor should he). So Baba Yaga doesn't seem to have much to do with what happened to the son.

And I don't understand what happened to the daughter. Did Baba Yaga kidnap her and give her to the brigands? Or did the brigands break into the castle, which the Baron couldn't do anything about because Baba Yaga had already killed all his guards? And when Elsa kills the hero in one hit, is that reflective of her great skill with the sword or Baba Yaga's curse?

Also what happened to the mother of these kids?

You can ask the castle guards about all these things. As I recall, Baba Yaga's curse was that the Baron would lose everything he held dear: his wife, his children, and (if not for you) all of Spielburg. So everything is considered her fault even if it didn't happen directly by her hand.