The gist of it is to build raised beds - 6" high - and NOT use your own dirt (which is full of weed seeds, or, if your dirt is like mine, consists entirely of weed seeds), but his special formula of peat moss, vermiculite and compost. The beds should be 4' x 4' (or longer, but no wider than 4'), and gridded off into individual 12" squares with lath or something. Into each 'square foot' you plant. One plant/seed for things like cabbage, 4 for smaller things like lettuce, 9 spinach or equivalent, and 16 radishes, carrots, etc. I've yet to get to the chapter where he explains how 6" depth will grow a 12" carrot (or perhaps you build a deeper box). I'm reading about how to build the just-right trellis for vining crops like tomatoes, even squash or melons (he claims to have grown a 35 lb pumpkin 8 feet in the air).
The obvious advantages: Easier. He says 80% of a typical row-garden isn't used for actual food-raising, but for paths and spacing, etc. You till up that 80%, but then you mash it back down, and grow weeds there instead. There should be "no weeds" in your SFG, and what does blow in and sprout are easily spotted and removed. You can grow more, because it's 'easy' to replant a square when you harvest it's contents.
Not so sure about: He writes for the family who wants to supplement their table for a few months. Not so much for someone who likes to fill the freezer with tomatoes. He recommends removing weeds/grass and laying down weed mat (not plastic) on which you build your box, but he has obviously never met my Evil Morning Glory. I'm toying with an idea he recommends to deal with gophers; give the 'box' a plywood floor, and set it up on cinder blocks or whatever. Hehe, that oughta stave off the Evil Weed... And make for an interesting view. I'm eager to read about how to plant carrots or potatoes or other typically 'deep' growing crops.
All in all, while I might not typically risk it, I might just do so this year. Our tax return is expected within a week or two, and if the other looming expenses can be ironed out, I might try to find room for appropriate materials to build a couple-few of these.
Also, the method has a website, if you'd like to see some examples...