I have tried the Back To Eden Style on one of my gardens for a couple years at which time it seemed to make sense.
It does make weeding easier and kept the soil moist, but almost too moist. Had some fungus issues every year with cucumbers and zucchini, tomatoes never really took off either where I was trying this method. I have come to the conclusion that the best way for me to garden is using raised beds. Anything I've tried in a raised bed grows great. I'm no expert either, just been vegetable gardening for about 4 years now. My first piece of advice for anyone just starting out at this point is go with raised beds if you can.
Few weeds, less need for water, and my plants seemed to do well. When I first started it - my soil was pretty clayful but the plants seemed to do well. I've since moved to an area with sandier soil, we'll see how my garden goes this year. I've had it mulched through the winter.
You don't till.
Tilling can disrupt all the microorganisms and worms and whatnot living underground. Add a barrier that will break down- mostly heavily moistened newspaper or cardboard- over the entire area you want your garden, then layer on compost, then layer on woodchips. You usually top dress amendments, but eventually, after a few years, you shouldn't have to amend the soil at all. The chips protect the plants from moisture loss so you water less and they decompose over time- adding fresh compost.
As rain falls, it pulls nutrients down with it, adding it to the soil. Additionally, weeds don't root well in loose chips, so you just rake them away. You just have to add fresh chips from time to time, year to year.
There are other similar methods.
Lasagna gardening is intermittent layers of this, that and the other directly on top of existing soil. I know less about it.