From an evolutionary standpoint (addressing what our kidneys and liver are "for"), our hominid ancestors didn't evolve to eat every single day. That just wasn't an option, and still isn't for wild things. Traditions of fasting arose in primitive religions long before we started eating highly processed foods.

Even if you eat reasonably clean, you're still inhaling toxins, ingesting them in your water (via additives, pollutants, naturally occurring contaminants and leaching from containers), and absorbing them through your skin. Reducing the load on your kidneys and liver for a day, or a few days, on a regular basis, might not be enough to eliminate everything that's built up in your body, but it's got to help.