How does the dreams and visions eventually become true?

In Book 20, Arthur and Mordred prepared to battle at Salisbury. The night before, Arthur dreamt that he was tied to a chair that was tied to a wheel, near a body of water filled with snakes, worms, and beasts. The chair turned over, plunging Arthur into the foul water, where he was overcome by the foul creatures within.

Later, the King dreamt again, this time of his nephew Gawaine. In the dream, Gawaine warned Arthur to avoid the battle at Salisbury, for Arthur would die if he fought his son. Gawaine told him to postpone the battle for a month, until Launcelot arrived with additional forces.

The next day, Arthur told his council of the dreams, and all agreed they should postpone the battle. Arthur sent two brothers, Sir Bedivere and Sir Lucan the Butler, to offer Mordred lands and goods in exchange for a month's peace. Mordred agreed to the terms, and was given all of Cornwall and Kent for it.

In Book 21, those dreams came true, as Arthur and Mordred prepared to meet to sign the treaty, but each distrusted the other and ordered his knights to be ready to attack if necessary. When they met in a tent, an adder bit a knight on the foot. When he pulled his sword out to kill the snake, the suspicious and anxious groups erupted into battle.

It was a brutal battle that left hundreds of thousands dead. When he surveyed the carnage, Arthur noted that only two Knights of the Round Table still survived: Sir Bedivere and Sir Lucan.

Arthur then saw Mordred leaning over a body of corpses, and prepared to attack him. Bedivere and Lucan reminded him of his dream, but Arthur was willing to take the risk, and he charged Mordred. He launched his spear through his son's body, but Morded struck his father on the head with his last strength before dying.

Arthur was so weak from the head wound that Bedivere and Lucan, who were gravely wounded themselves, could hardly hold him long enough to get him to a nearby chapel. Suddenly, a great noise arose on the battlefield, and Lucan saw scavengers pilfering the dead of their money and jewels. The brothers decided to take Arthur away from there, but the King was too weak. In attempting to lift Arthur, Lucan aggravated his stomach wound, which caused his guts to fall out, at which point he died.

Knowing he was near death, Arthur asked Bedivere to throw his sword, Excalibur, into the water and then to report what he saw.