The liquid never comes in contact with the laundry. By dropping them into your washer instead of using laundry detergent, the manufacturers claim that they'll get your clothes clean. It's important to note that the manufacturers of these devices seem to provide much more accurate information than the distributors. The distributors tend to leave out the important bits, about which more below. It has nothing to do with quantum mechanics, or activated water, or mysterious magnets, or anything else of that sort.
Laundry detergents consist of a number of components, the largest of which is very much like soap.
Soap molecules are long chains of atoms with a special attachment at one end. The lye molecules attach themselves to the ends of the fat molecules, and the result, when the mixture cooled, was soap. So when you put greasy clothes in a soap solution, the oily ends of the soap molecules attract and hold the oil and grease in the clothes, and non-oily ends attract the water. This allows the oil to be pulled out and washed away. But in today's world, unless you have a profession that puts you in contact with a lot of oily things, most of the soil in your clothes consists of particulates like dust and dirt, or water-soluble chemicals from sweat, and so on.