Soap (and detergent) reduces the relatively high surface tension of water and so enables the water molecules to penetrate better the pores of the food residue clinging to the dish. Unless that happens, some of the food particles - although surrounded by a sinkful of water - can remain dry, hard and difficult to remove.
Soap helps clean dirty dishes in another way. Oil and water molecules naturally repel each other, and thus the water alone cannot effectively penetrate oil, or oil-coated food, on the dish. However, with the aid of an emulsifying agent like soap or detergent, the water and oil will mix and travel down the drain together.
A soap molecule is an emulsifier because of its split personality. One of its two ends is hydrophilic. It is drawn to water and shuns oil. The other end behaves in the opposite manner. It is hydrophobic - repulsed by water and attracted to oil. The soap molecule's hydrophilic end binds itself to the water, while the hydrophobic end attaches itself to the oil. A little elbow grease on your part, and the oil molecule glides off the plate. The oil-soap-water emulsion is not, however, permanent - as you can tell from the oil layer that sometimes lightly coats your hands.
In addition to its molecular properties, soap makes the water more viscous by its sheer physical presence. This increase allows the fluid to carry more and larger particles, so that freed oil floating in even relatively dirty water usually is not forced to take up residence on a soaking plate once again.
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