How would one connect multiple PCB's together for compact designs?
Well despite the fact that I didn't feel like watching that intro video :)  Looks like they're just stacking.  You start by looking at your available surface area and volume.   Can you fit everything on one double sided board in your given surface area?

No well do you have more height?  You could make another board and stack it on top using connectors or wires.  Now you have to think about the height of the components and how these two puzzle pieces would fit together.  What goes on what board, and you have to think about how this will affect the thermals.  

You could put a bunch of boards at 90 degree angles too, doesn't have to be just stacked.

Can't go up? Maybe switch to double sided with blind buried vias.  A little more expensive and more complex to design but you can treat both sides of the board as their own space.  In other words that big bga on the top doesn't have to take up all of the bottom as well.

Still not fitting?  Hmm well time to start looking at the components.  Can you buy known good die for your processor and do die on board?  More expensive, and you either need a good volume relationship with the provider, or maybe you're making the ASIC yourself.

Or for things like cell phones or other consumer electronics you can use PoP (package on package).  Where the memory parts are made to sit on top of the processor:



Got a lot of money and control over the your parts?  There's always stacked die all in the same package:



Of course there's just general miniaturization too, using the smallest packages, pushing for smaller spacing on the PCB, etc. In general the further down the list you go the more it costs, and the more risk.  But you can combine any of these techniques depending on where you want to go.

That's all I can think of for now, maybe someone else can add a few more.