One reason is to tap into the man-power of the Balkans, and a soldier who converted and had little to no knowledge of his previous family is a formidable soldier for they dont have as much to lose. Its also the same reason the Byzantine emperors wanted to use non-Greeks as their personal bodyguards or army (Varangian guard). Foreign boys would not be loyal to a certain nobles that could attempt overthrowing the Emperor or Sultan. Instead the foreign bodyguard would more likely be loyal to the Sultan since they did not know of the other nobles.

Thats why the Devsirme system took foreign or non-Muslim boys as a blood tax and trained them for war. There is also a conception that boys that grew up in cities were softer than boys who worked on farmland.