The people in that blog have some respectable carpentry/DIY skills. The vertical pieces could have been attached in any number of ways. There's a really thick shelf about 2/3 of the way up, I'm assuming that this is where most of the attachment and stabilization force is situated. I'm pretty confident about that especially because all of the other shelves seem to be adjustable shelves and are therefor not structural. A person could use pocket holes (using something like a Kreg jig) or L brackets to make a more solid connection to the wall, countertop, or whatever else, but I wouldn't rely just on those for shelves intended to hold canned food and small appliances. It's really too bad there are no close-up pictures in that post to see how the joinery was likely done.

Some of the shelves in that photo appear to have solid wood fronts (the thicker looking ones.) In our shop, we call this style of shelf a "book shelf", where most cabinet shelves (fixed or adjustable) are usually some sort of plywood material with edge-banding. Book shelves usually have a solid wood front attached using a plate jointer or with dowels. The solid wood front is then sanded and profiled. Solid wood fronts add the visual effect of making the shelf look thicker and add to the structural integrity of the shelf. In cases where a person would expect the shelf to span a longer distance and/or hold heavier objects sometimes a solid wood back is applied as well!

I hope at least some of that was helpful!