What exactly happens while washing clothes
When you wash a piece of cloth, even without soap, it's better to agitate. The vibration, friction, etc. will help speed up the process of mixing the dirt/grease with water. 

Even the very definition of "agitation" uses the example of washing. From Vocabulary.com 


  Agitation is the act of stirring things up, like the agitation of a washing machine that moves the water, detergent, and clothes around and gets the dirt out


And as you rightly pointed out from another question, "the froth has little or no effect on the detergent action".