How did they make the nice moving shots in this video? https://prismic.io/ - has a very nice explainer video and I was wondering how they made the nice "tilted" sliding effect on the screencast video? Also... how are they making the video being focused on the center, but blurry on the sides?

I know it's probably a bit of a noob question, but I'm still relatively new to AE.
It's two very basic tools in After Effects.  The first is simply either a perspective correction or a 3d transform depending on which way you want to do it coupled with an x/y movement across the frame.

There are also a few different ways you can approach the blur such as using a layer as the basis of the amount of blur and using a mask or using a filter that supports a blur that changes with the distance from center (I think there is a built in filter to do this under the blur filters, but if not, I'm sure there are third party ones.)