How does soap actually remove grease and dirt?

Soap molecules have a dual structure: one end is hydrophilic (attracted to water) and the other end is hydrophobic (attracted to oils and fats). When you wash with soap and water, the hydrophobic tails of soap molecules embed themselves into grease and oil particles, surrounding them and forming structures called micelles. The hydrophilic heads face outward toward the water. This arrangement lifts the grease away from the surface being cleaned and suspends it in the water, since the outer shell of the micelle is water-compatible. When you rinse, the water carries these micelles away along with the trapped dirt and grease. Mechanical action like scrubbing helps break up grease deposits and speeds up the process.