I'm no architect, but I think the cost savings as compared to modern techniques (but very much, if not exactly, like older techniques) is the avoidance of 1) clearing and leveling the land and 2) pouring a huge reinforced concrete slab.
Before the advent of hydraulic bulldozers it was just too cost prohibitive to clear land and pour big slabs for mass-scale housing. Instead, you would pour (or build) some small footings, and then lay some beams on the footings, possibly elevated by some short columns (or simply stones or concrete blocks). But unlike what would be common for cheap housing 50-100 years ago, building codes won't allow you to build a whole house on simple footings like that because of the potential for ground instability exacerbated by uneven weight distribution and uneven weight-bearing capacity of the soil under each footing.
But what you can do, apparently, is use "cantilevered" footings tied to a small number of deep, robust, secure pads. Those pads are what guarantee the house won't move, and all you need are some shovels and a pick-up truck, and you don't even need to level any land. You then "cantilever" some simple concrete footings off those pads using reinforced concrete beams (which I bet are probably flush with the ground and look like a gratuitous use of concrete). I'd bet money that such cantilevered footings are what are underneath many or most of those columns you see under those houses, the rest of the columns sitting directly atop the pads.
It signals another example of changing practices in this century due to new ideas around materials accompanied by precision machine crafting. Cross-laminated timber, for example, is very hyped now since it can withstand huge loads and charring while still being a cheap and familiar "wood, glue, preservatives and pressure" composite, capable of being shaped into panels, arches and doorframes out of the factory with a CNC machine for reduced build times. Yesterday I saw a video of a CLT passive house built in Australia: The build, while not aiming for cost, was made simpler in design by having the load bearing parts of the structure also handle most of the insulation, and it put up impressive test numbers.
Unfortunately, I can't find any technical material on the school's website. Here's the page for the various homes: http://ruralstudio.org/project_tags/20k/. Notably, some of them do seem to use modern concrete slab foundations, and at least one of the pictures shows a bulldozer in the background. Looks like they have a bunch of different designs.
[1] Basically just like people build decks today. Of course, lots of decks are dangerous, shifting and collapsing, precisely because the techniques aren't very forgiving of mistakes or poor siting.
EDIT: Here's a blog post with a better description, which seems to be discussing the same house(s) depicted in the Fast Company article: "Siteworks are a major cost item and cantilevering the floor joists past the foundation piles reduces the area of siteworks. ... Cantilevered floor girders act more efficiently as beams, making smaller and less expensive timber sections sufficient. Floor joists are cantilevered from the girders for the same reason." https://misfitsarchitecture.com/2016/08/31/architecture-misf... So it seems the usage of "foundation" in the other article was misleading. And these definitely aren't new techniques, not if you count the stock of 50-plus-year-old homes built in poor, rural America (i.e. not simply the stereotypical New England farm house). So presumably what's innovative is that the students have done the proper analysis and testing to refine them, ensuring they behave as intended as well as meeting modern standards of safety and reliability.
