In that video at 1:05 it looks like an in camera effect. IOW, the guy holding the camera couldn't keep his hand off the zoom and couldn't hold still. If that sounds like a criticism, it may be. That kind of shooting was boring to me in 1981 when every tv series from Hill Street Blues to whatever. A very little of that kind of thing goes a very long way.
If you haven't got it in camera you can fake it in AE. The easiest way would be to take just the clip you want to shake, zoom and rack focus and animate scale, position and animate a box blur. Just turn on motion blur and jack the shutter angle in the Advanced Composition settings up to about 720º to exagerate the effect. You could even spice it up more if you enable time remapping, throw in a few extra keyframes and then drag them around a bit.
If you don't know anything about setting keyframes then start with the basics by going through at least the materials you'll find at the Getting Started with AE link on the forum home page. I'm not going to post a link because I've done it a million times and only a few bother to click on it.
Here's an example shot with my iPhone camera while I was writing this post:
Spent about 2 mins on the project. As you can see it's not nearly as organic as an in camera version of the same idea.
As Rick says, everything you're seeing in this clip is done in camera, except for the text.
The slow mo stuff is shot with a high frame rate camera. Plenty of DSLR cameras will shoot high frame rate these days.
The action stuff is shot with a high shutter speed. This reduces the level of motion blur captured by the camera, giving it that edgy, pristine look that was made famous in films like Saving Private Ryan. Most mid-level video cameras and DSLRs will allow you to set a high enough shutter speed to get this effect.