In "Star Wars: Episode III – Revenge of the Sith," Darth Vader, formerly Anakin Skywalker, learns about the tragic fate of his children—Luke and Leia—mainly through the machinations of the Emperor and the choices made during his fall to the dark side. After Padmé Amidala's death, Anakin believes that she betrayed him by having these children and, in his anger, he lashes out against Obi-Wan Kenobi when he realizes that he has been manipulated.
When Obi-Wan confronts Anakin on Mustafar, he is trying to bring him back from the dark side. Their duel represents the final breaking point between them, illustrating how far Anakin has fallen. After defeating Anakin, Obi-Wan separates the children to protect them from Vader, hiding them from the Emperor.
Darth Vader's knowledge of his children is limited at first, but as the saga progresses in the original trilogy, he begins to sense their presence. It's not until "The Empire Strikes Back" that he fully realizes that Luke is his son. The conflict between Anakin's love for his children and his allegiance to the Sith creates a deep internal struggle that ultimately leads to his redemption in "Return of the Jedi."
In summary, Vader's awareness of his children's existence is revealed later in the timeline, and the facts surrounding their separation are rooted in both Obi-Wan's protective instincts and the overarching plot to keep the Jedi legacy alive amidst the Empire's reign.