From what I understand, each boss in this crackdown event gives you Clout when you defeat it. On the left of your screen you'll see an overall Clout for the stage. Each time you recieve clout it reduces the stage clout. Once you have reduced all of the stage's clout, you progress to the next boss stage.
As for Clout itself, it is not a resource used in game (from what I can tell). Hope that helps.