Yeah, any guess is as good as mine. The first time it happened with me though, it wasn't like this video at all. I had an Assault Juggernaut with the care package, I stayed alive and my first kill with the Jugg earned my Predator Missile. I was so excited, I let two enemies overwhelm me and I died before the Jugg could earn my final killstreak.

It's definitely some kind of glitch, and the game sometimes thinks your care package is your predefined killstreak. I'm trying to see if I can replicate this glitch, by earning the Predator Missile and then going for the care package, as shown in the video.