What exactly happens while washing clothes I am reading about soaps , micelles and all that. I understand how soap behaves in water contaminated by oil. Soap helps in emulsifying the oil into water.

Let us suppose now we have dirt(i.e oil, grease etc) not alone in water but present on a some cloth. Why can't we put the cloth just in water to get it clean ? What precisely is the need for agitation in a washing machine or by any other method ? 

I searched and found that agitation produces foam but according to this post foam does not alter the washing process anyhow.
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".