Edited some time later. Probably a good option for that kind of video is using PowerPoint, which have some transitions that can be used for animation, and also have some built in animation tools.
Libre Office have more rudimentary transitions. But it is also an option because that video have very simple ones.
But if you are looking for a more specific animation software:
Some free options I know.
Express Animate. http://www.nchsoftware.com/animation/index.html It is easy to use, can export to video and html5 canvas, can animate, rotate, scale, and you can edit the velocity and trajectory with ease in - out nodes. Very lightweight and it is free.
Blender http://www.blender.org/, you can do a 2D animation just using a top view for example. There will be plenty of tools you won't use... but it's free.
HitFilm Express. A fully functional compositor (Think of after effects) https://hitfilm.com/express
Another option, flash like and free: http://www.synfig.org/cms/
If you like a free stick figures tool: http://pivotanimator.net/.
And other not too expensive ones.
A good payed option is Crazy Talk Animator http://www.reallusion.com/crazytalk/animator/default.aspx it is not expensive, and it is verey flexible.
Another is Anime Studio (Now Moho) debut http://my.smithmicro.com/anime-studio-debut-10.html. Probably more specialized than Crazy talk.
In some cases you will need to syinc your voice trial and error, you can use http://audacity.sourceforge.net/ to record.