There are several ways to do this. The most common term for this effect is "cloning". Search YouTube for "cloning vegas movie studio" and you'll find lots of examples. However, that search will include examples done in the full/pro version of Vegas, which has the bezier mask feature that makes this kind of thing very easy.

First of all, when shooting your clone videos, the camera must be on a tripod, totally locked down. One common mistake people make is to move the camera ever so slightly when starting or stopping recording. You don't want your camera to move even a single pixel's worth over multiple shots. If your camera came with a little infrared remote control, now is the time to dig it out so you can start and stop recording without touching the camera at all! (Or, just let the camera run continuously without stopping in between takes.)

Secondly, lock the exposure on your camera before you start recording. That is why the Tin Whistle guy had to adjust his videos to get them to match -- the exposure wasn't locked. (Or else the lighting in his room changed over time.)

CLONING USING A MULTIPLY MASK (Not the easiest way)
You could use white-black gradients like the Tin Whistle demonstrates, but to get 3 clones on the screen you would need a total of 5 tracks, and VMS only has 4.

To do it this way, go to the Media Generators tab and select the "Linear Black to White" gradient and drag it to your top track. Now open up the FX dialog for the gradient event and click on the #2 control point. Change the color for this point from black to transparent by dragging the triangle (below the word "Color") all the way down to the bottom of the checkerboard. Now, in the track header at the left edge of this track, go to the icon to the right of the slider and click it to change from "Source Alpha" to "Multiply".

Next, drag one of your clone videos to the track directly below the gradient. On that track (the one with the video) go to the track header at the left and click the "Make Compositing Child" icon. This is the second icon to the right of the slider and looks like a skinny rectangle with an arrow pointing down. It changes to pointing up after you click it, indicating that now your video track is a child to the track above it. At this point, any place that has solid white in your gradient mask will show your video 100% solid, and anyplace that is checkerboard in the mask will change your video to transparent. Where you have a gradient, your video will become gradually transparent.

Now, put your background clone video on the next lower track. You should now see both clones.

This Multiply Mask approach is a little awkward, but the benefit is that instead of using a color gradient media generator, you could create your own mask in photoshop or even MS Paint and create a mask in any shape you need. If you cannot create a transparent background in your photo editing program, just create a black and white mask, then apply the "Mask Generator" effect to it (from the Video FX tab).

CLONING USING COOKIE CUTTER (Easy way)
The only problem with this approach is that VMS has a very limited set of shapes for the cookie cutter. The benefit is you only need one track for each clone, so you could have up to four clones in VMS.

Put the leftmost clone on the top track. From the Video FX tab, apply the Cookie Cutter effect. Momentarily set the border color to red with a size of 0.02 or so, so you can see it, then set the cookie cutter shape to a rectangle or square (or whatever shape works best for you) and adjust until you are keeping just the amount of clone you need from this track -- i.e. the left 1/3 of the video. Change the border back to 0.00 to get rid of it.

Now put the middle clone on the next lower track. Apply the Cookie Cutter effect and adjust as before. Note that you don't have to isolate your middle clone -- you can keep the left 2/3rds of the image because the left 1/3 will be covered up by the top track.

Finally, put your full background clone on the next lower track. No cookie cutter is needed for this track.

The best advice is to just play around a lot and experiment on a "test" video before you sit down and try it on the "real" project.

Tim L