As mention in the question, the car was moving then every object inside the car was also in the motion state with the same velocity as the car had. when the brake applied on the car, the brake only stopped the motion of the car than the objects inside it. According to the first law of motion, due to inertia, the body keeps its state intact until an external force applies to change its state. Because of inertia, the coffee inside the cup was in moving state with the car. When brake applied the coffee spilled out of the car to maintain its state and spread all around. If the cup would keep out of the cup holder, It will move also forward on the applied brake.
