She agrees because they've had a long affair
Igraine
Igraine's first husband was Gorlois. Her second husband was Uther Pendragon.
Merlin gave him a drug that transformed Uther into Gorlois's likeness. Uther then lied to Igraine about how he'd slipped from the seige to see she was safe
According to most sources, King Arthur was born of King Uther Pendragon and Igraine, who was wife of the Duke of Cornwall before she married Pendragon. Uther, consumed with obsession/love/lust (whatever you believe), rallied his army and attacked Cornwall. In the course of the battle, Cornwall was killed, and Igraine and Pendragon married, bearing Arthur. However, threat of Saxon attack moved Pendragon to foster out his child. Arthur grew up never knowing of his royal birth. Until, of course, he first laid eyes on glimmering Excalibur.
Uther Pendragon, the King of England, was deeply in love with Igraine of Cornwall. Uther's assistant, Merlin, a magician, promised that they would be married if Uther gave Merlin a son. So, Uther and Igraine fell in love and had Arthur who was sent to live with Sir Ector for safety and because Uther was going to die.
King Uther was crazy in love with the Duke of Cornwall's wife, Igraine. He wanted her as a lover but she refused him, loyal to her husband. King Uther became depressed. Merlin promised Uther he could spend the night with her, disguised as Gorlois (the Duke). Merlin's only condition was to take their child once he was born. Uther, desperate for love with Igraine, agreed. The Duke was killed in Terabyll. And Uther & Igraine ended up together. They had their son, and gave him up to Merlin, as promised.
King Arthur was the son of Uther Pendragon and Igraine.
Arthur's father was King Uther Pendragon and his mother was Igraine. Igraine was the queen of the Cornish king, but Uther was enamoured of her and fooled her into thinking he was her husband by the use of Merlin's magic, and she conceived Arthur.
Igraine sent her there to learn and grow into a proper woman against Uther's wishes. Uther wanted to kill Morgan instead, erasing his previous marriage. He also had her mother killed to marry Igraine.
King Uther Pendragon and Lady Igraine of Cornwall
Igraine (Ygraine) of Cornwall. She was seduced by Uther Pendragon disguised as her husband Gorlois.
Igraine (Ygraine) of Cornwall. She was seduced by Uther Pendragon disguised as her husband Gorlois.