What’s the process that makes it good at something so far off from its composition?
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Water is polar, so a part of its molecule is charged positive, and other part is charged negative. Oils and fats are non-polar, there is no charge difference across the molecule. Polar solvents dissolve polar molecules, and nonpolar solvents dissolve nonpolar molecules. Soap is made from fats, but during the saponification reaction fat molecules are modified: before the reaction they look like a pitchfork, with three long molecules joined at one base. Adding a base like NaOH breaks teeth off that pitchfork, with Hydrogen attaching to the pitchfork base making it into glycerin, and Na and O attach to cut off ends of pitchfork teeth, turning them into and interesting molecule: molecule body itself still has no charge, but the end with oxygen and sodium attached has charge. So now we have an “adapter” molecule: neutral part interacts well with nonpolar substances, and charged end interacts well with water or other polar solvents. Now water can interact with grease through this molecule and dissolve things that were not soluble before
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