The origins of the devşirme are shrouded in mystery because of the lack of contemporary records, so historians can only guess at how and why it came about the way it did. It's generally agreed that the sultans first started establishing a household of followers by levying a tax of 1/5th on all slaves transported from the Balkans to Anatolia, where they could be assessed at Gallipoli. But this system was inherently unreliable, the number of slaves taken would not be stable every year. So the devşirme began, which was a more consistent system of acquiring manpower. But the logic for determining the who, where, and how of the boys' conscription? No one really knows. One historian has argued that it was a tax imposed only on villages which did not voluntarily surrender at the time of the initial conquest (V. L. Menage, "Some notes on the 'Devşirme'", Bulletin of the School of Oriental and African Studies 29 (1966)).

As for its purpose: the most commonly accepted explanation is that the Ottomans wanted to recruit individuals who were not connected in any way to the old Muslim/Turkish aristocracy. Christian boys wouldn't be totally disconnected from their roots, but they would generally not have any link to the rest of the empire's elite, and would thus be more loyal toward the sultan. So in that sense it wasn't about controlling the Christians, it was about controlling the Muslims.

Also to expand on /u/Chamboz's answer, the devshirme strongly resembles other slave-soldier systems, such as that of the Mamelukes, who still "recruited" from among Christian slave populations even when they didn't live in the empire. As far as I'm aware the Mamelukes never enslaved Egyptian Copts; if they did it's not something I have ever come across.

This line from /u/Chamboz more or less includes the logic of all such systems:

wanted to recruit individuals who were not connected in any way to the old Muslim/Turkish aristocracy. Christian boys wouldn't be totally disconnected from their roots, but they would generally not have any link to the rest of the empire's elite, and would thus be more loyal toward the sultan.

There was also in theory a stricture against taking Muslims as slaves.

I would give a read to Patricia Crone's Slaves on Horses.