How did the video editor for this clip make part of an image transparent and let a moving image play from behind?
probably with masks. see here:
Fisst step, cut holes in your photographs using Photoshop or by bringing the photos into a non linear editor like Premiere Pro or many others that supports masks, or - The hardest way to learn an the slowest way to work for a newbie - Bring your photos into After Effects and cut holes in them in there with masks, track mattes or, and we're getting a little crazy here - referencing another layer in the project, something like Set Matte.
Second step, use an app that allows you to stack layers if videos and images like iMovie, Premiere Pro, I think even Movie maker, or if you are up for taking a couple of days to learn the interface and the techniques, After Effects and put your still images on top of the video.
There are even smart phone apps that will let you cut holes in your photos and put videos behind the images. If you want to learn to use After Effects efficiently then you'll need to spend some serious time learning about a bunch of things but you'll be able to create any composite you have ever seen in any movie or do any special effect you can imagine. AE is not always the right tool or the fastest tool for the job but with the right training and the right plug-ins you can do must about anything you have ever seen in AE.
To learn more about AE start here:Basic AE