How does soap work in chemistry?

Hint: We need to know that Almost all compounds fall under two categories, Hydrophobic and hydrophilic. Out of which Hydrophilic is water loving whereas hydrophobic is water hating. When water and oil mixed together they got separated into layers likewise hydrophobic and hydrophilic species do not mix with each other.

Complete answer:
We need to know that the soaps are mixtures of sodium or potassium salts of fatty acids which can be derived from oils or fats by reacting them with an alkali (such as sodium or potassium hydroxide) at 80°–100 °C in a process known as saponification.
\[fat + NaOH \to glycerol + {\text{Sodium salt of fatty acids}}\]
The cleansing action of soap is determined by its polar and non-polar structures in conjunction with an application of solubility principles. The long hydrocarbon chain is nonpolar and hydrophobic (repelled by water). The "salt" end of the soap molecule is ionic and hydrophilic (water soluble). When grease or oil (non-polar hydrocarbons) is mixed with a soap- water solution, the soap molecules work as a bridge between polar water molecules and nonpolar oil molecules. Since soap molecules have both properties of nonpolar and polar molecules the soap can act as an emulsifier. An emulsifier is capable of dispersing one liquid into another immiscible liquid. Micelles are formed in soaps having both hydrophobic and hydrophilic parts.

Note:
We need to know that the critical micelle concentration can be called the concentration of surfactants at which micelles formed. As soon as CMC reaches, surface tension becomes constant or becomes relatively lower. And before CMC, it is strongly equal to concentration of surfactants. Micelle can be formed by aggregating the molecules together.