I stay at a rented place and there is no washing machine. I, now, wash clothes in a bucket. When I add detergent, sometimes there is froth, sometimes there is not. I wonder if this had any impact on the washing process. Though I am aware of micelles and all that, but I wonder if froth formation is nearly redundant.
The froth has little or no effect on the detergent action. In fact detergent manufacturers have to add anti-foaming agents to stop excessive foam generation in automatic washing machines.
Froth/foam is generated because surfactants in the detergent adsorb at the air water interface and stop the water film that makes up the walls of the bubbles from collapsing. Note that the surfactant does not cause bubbles to form - bubble formation is always energetically disfavoured and the foam is only kinetically stabilised. The foam is caused by agitation entraining air into the surfactant solution so the amount of foam will depend on the intensity of the agitation as well as the type and concentration of the surfactant.
It is conceivable that in some cases the foam may assist with dispersal of mineral soils. This is the basis of the industrial froth flotation process. However this plays at most a very minor role in domestic washing machines. Not that it's directly relevant, but I mention it out of interest, foam is thought to play a role in cleaning teeth i.e. in toothpaste. It entrains particles of food so they can be spat out along with foam.
I like the technical answer. Here's a practical one: Froth indicates that you have more than enough detergent to do the job. Use too little detergent for an especially oily mess and all of the available detergent is "consumed" by the oil with none left over to make froth. That's why after using dishwater for a while, the plates don't "feel" clean anymore. One might say that any froth is a waste of detergent for cleaning purposes, but others say just a little froth is a small price to pay for the reassurance it gives. A lot of froth is simply a lot of waste.