I'm relatively sure most video editing programs above the freebie that comes with a video camera can produce that. I'm 100% sure that could be created in Final Cut Pro, Avid, Premiere Pro, Sony Vegas and even the lighter version of several of these programs. Although, I suspect many of the effects used were presets, most of what was done could easily be done with standard tools. Most of it is just track layers being re-sized and repositioned. I doubt this was done in After Effects or another designated composting program, but it could be. My advice would be to use whatever editing program you are most comfortable with and learn either the preset effects similar in that program or better yet, learn how to create each effect. The tools/ effects/processes you will need to learn are (I am being very general so that it will apply to many editing programs) are:
1) Resizing track layers so that you can have one video on top of another.
2) Manipulating opacity/transparency
3) Basic track positioning and animation via keyframes
4) Knowledge of Cropping and Masking will be a big help too.
5) Familiarity with your arsenal of transitions built into your editing software.
6) The creativity to come up somethign much better than the video you posted 