Rod,
There are a couple of ways to do this...and this isn't the only one.
On the software side, programs like Adobe Premiere Pro, Avid Media Composer and Apple's Final Cut pro all can do a 'picture in picture' effect - where you have a second track of video that you've shrunk down and put anywhere on a screen (such as a corner).
You capture the youtube file (which will go in the editorial software- you show it and record people's reaction with a single camera.
All that's left is to put it together.
You put the People reacting on the first track of video.
On the 2nd track (with a picture and picture effect) you shrink down and show the youtube video.
Now - the only thing we need is to sync the youtube video - it's pretty easy to get close when you first put the video in your editorial software - you'll be less than a second or two off - after all, you can hear both the original music and the youtube video. Then you just slide it left or right to get it into sync.
The more professional way would be to grab the youtube video add a 'beep' before it, and use the beep to figure out it's placement in editorial software to sync it. IN many ways, this is like old school film - where you shot film (with no sound, but with a clapper) and you recorded audio (with no picture, but the sound of the clapper.) Line them both up and they're in sync!
Let me know if you have further questions.