You will find a purple progress bar in the upper-right corner of the playing screen. Clients will give you a specific amount of reputation points after a service, depending on the haircut and/or the beard trim they choose. Once the progress bar is full (E.g. 40/40), the star will turn golden, and a new story will be available! 

Tap on the bubble speech icon as soon as it appears to advance in the game.
There are some objects in your barbershop that will increase the number of reputation points you get. Pay attention to the object descriptions and invest your in-game money wisely!

Your rep points will also help you unlock (and upgrade with purple stars) new hairstyles and new beard trims. Tap on the scissors icon to know which one's next!