The kinetic theory of matter or particle theory describes that all matter consists of many, minute sized particles which are constantly moving or in a continual state of motion. The movement of the particles is determined by the amount of energy they have and their relationship to other particles. The particles can be molecules, ions or atoms.
In this situation, it is detergent molecules. In a hot water, the detergent molecules will gain more energy from the heat. Thus, it will move rapidly and bumps to other molecules more frequently. This increases the solubility of the detergent in the water. Changes in temperature may increase or decrease the solubility of a certain substance in water, depending on the substance properties. Same goes for the dirt on your clothes. The higher temperature will make the dirt loosen (by breaking bonds from the clothes)
This means that the detergent/water solution will be able to dissolve (form bonds with) the dirt faster and better, because the bonds between the detergent and water and the bonds between the detergent and dirt will form faster and will be strong enough to prevent the dirt from reforming the bonds with the clothes.
These bonds will also break faster, but the presence of a detergent helps to stabilize the solution and keep the dirt dissolved in the water. Once the dirt is dissolved in the water, the clothes will be clean, and the dirty water can be easily discarded.
In summary, hot water helps the cleaning because it can increase the solubility rate for the detergent to dissolve the dirt.
