This question was actually answered a little while ago on Scienceline here:

The answer borrows heavily from this online article:

The long short of the science of stain removers is that they can remove stains in four main ways (dissolving, emulsifying, reacting, and masking) and most cleaners use a combination of these. Clorox’s website has a great interactive page on how each of the ingredients in their products work.

If you look under “What’s in it anyway?” you can click on each of the ingredients individually and read what they do.

Try to label each ingredient as either a solvent (something that dissolves things), emulsifier, enzyme/reactant, or masking agent!

Here’s another example of how the ingredients in a famous stain remover work.

OxiClean Versatile Stain Remover (Powder) Sodium Carbonate – Water softener that prevents metal ions in laundry machines from interacting with detergent

Sodium Percarbonate – This can be thought of as “solid hydrogen peroxide”. When mixed with water, this substance produces the oxidizer hydrogen peroxide. Hydrogen peroxide (H₂O₂) readily reacts with the organic molecules that give stains their colors, thus rendering them broken and colorless.

Ethoxylated Alcohol C₁₂-C₁₆ – This is a surfactant, like soap. It helps to emulsify fatty stains.

Sodium polycarboxylate – Prevents dirt and stains from depositing back onto clothes.

Sodium metasilicate – I’m a little unclear about what this does, but it seems like it helps to emulsify and to prevent solids from depositing back on to clothing.

Ghost of BILLY MAYS – Scares stain molecules out of clothing. (Just kidding!)

Interesting question! Stain removers typically employ one or more of several mechanisms to remove stains. Stains are the result of an interaction between two chemically or physically dissimilar objects/materials. One common way to remove stains is by using detergents. Detergents usually have emulsifiers or surfactants that make it easier to remove the stain by coating the stain molecules and making it easier to lift them from the stained material. Another way to remove stains is by dissolving them in a chemically similar substance. Other methods include using enzymes to "digest" the stain. Another way yet is to simply "hide" the stain (e.g. bleach oxidizes the stain molecules so they appear lighter and less visible on a white shirt).