Briefly, the primary attribute of a clean label is that the ingredients are declared in simple, recognizable terms, without any chemical-sounding names or other implications that the ingredients are less than wholesome, pure foodstuffs. This is a laudable goal, provided it is done in accordance with applicable regulations. For example, declaring beef meal or beef by-products as "beef" for the sake of simplicity, or failing to disclose the presence of substances that are not clearly exempt from declaration under the law because they may be unappealing to the purchaser, does not make for a clean label. Rather, it is just a false and misleading one.
About the Clean Label Project
There is much more to the clean label concept. Regardless, the Clean Label Project (cleanlabelproject.org) does not seem to have anything to do with the "label" at all. Rather, its stated mission is to "educate the consumer so they can make informed choices on cleaner options every time they shop." This is accomplished by conducting a battery of analyses for toxins and other potential contaminants in hundreds of commercial pet foods.