I'm assuming they were using it to keep weeds down. You cut holes in it where you put your plants so that there are no weeds in between the plantings. That alone wasn't a completely terrible idea, but there was no way that a garden that small would made enough produce for more than a couple of people. Even then, it would be months before anything was edible.

Ideally, you would put the cardboard down where you want to plant, wet it, then let it kill off the weeds/grass and naturally compost in the soil over the fall/winter so that it's ready for spring planting.