The fibers attach themselves to each dirt speck with van van der Waals forces. Working as a team, many fibers apply powerful enough forces to dislodge the dirt (loosened with a small amount of water) and carry it away, leaving the surface naturally dirt-free. This is new-fangled cleaning with physics using nothing but the adhesive power of forces—cleaning mechanically and without chemicals. The dirt stays locked inside the cloth’s fibers until you wash it in hot water, which makes the fibers uncurl slightly and release their dirty content.
If you use a microfiber cloth, there’s no detergent involved whatsoever, so how is the dirt removed? Instead of detergent, we rely on millions more fibers that can sweep dirt away, mechanically and physically removing it from the surface.