Saul was asked to destroy the Amalekites because of what they did to the Israelites when they left Egypt. The Amalekites attacked the Israelites from the rear, targeting the weak, elderly and those who were lagging behind. This showed a lack of fear and respect for God, who had protected the Israelites from the Egyptians. As a result, God commanded Saul to destroy the Amalekites and all their possessions as a form of punishment for their actions.