Why did private landowners need to build armies in Feudalism and Manor Life

Because of the way Feudalism works. It consists of a contract between a lord and his vassals, or lieges, where each liege pledges to serve the lord with his work, and in return the lord promises to keep them safe from outsiders. So the landlords, who owned most of the economic power during the Middle Age, bought soldiers and built armies to protect the land they owned, and therefore their vassals.