A grease/oil spot of about a milligram would make a droplet about 1 mm in diameter. This is too large to stay suspended in water - it would float to the surface, where it would combine with any other tiny droplets, even if these droplets were coated with surfactant. When you withdraw the clothing from the dirty water, or drain the water thru the fabric, the oil/grease would be trapped again on the fabric.
However, if you agitate all the 1 mm droplets, each covered with a monolayer of surfactant, in the presence of more surfactant in the water, the droplets will deform, elongate, and split, generating more surface area (the extra surfactant molecules in the water will rush to coat the oil/grease). As agitation continues, the tiny droplets become smaller and smaller, always covered with a monolayer of surfactant, because you put enough into the water at the beginning of the washing cycle.
Finally, the droplets become small enough that their buoyancy is negligible compared to Brownian motion (maybe 0.005 mm or less) and they act like large dissolved molecules. At this size, the particles are very difficult to filter out of the water using ordinary fabric, which is good, because you don't want to trap them, you want to drain them away.
One way to make an emulsion of small particle size is to add an organic acid (e.g., oleic) to an oil (e.g., canola), then pour this into a sodium hydroxide solution. The acid saponifies in intimate contact with the oil and the oil disperses into small droplets coated with oleate anions. Sodium ions are also in solution. This happens spontaneously thru a chemical reaction.
However, if you drop canola oil into a sodium oleate (a surfactant) solution, it floats without significant emulsion forming. This is typical of ordinary oil/grease - you don't have a readily available chemical reaction to form small droplets. But if you shake it you break the oil into tiny droplets, and you will get an emulsion that is stable (i.e., doesn't separate into two layers) for a while. The emulsion will be stable longer if the droplets are smaller. You have substituted mechanical energy for the chemical energy, and the surfactant acts to maintain the stability.
If you have insufficient surfactant, the droplets won't get small enough to become stable in water. If you use excess surfactant, you will get smaller particles and a more stable emulsion, and lots of foam.