Think of one individual blade. Like all mass, it has inertia and so is actually trying to travel in a straight line relative to the universe. However, the hinged "tie" to the rotor mast means that it rotates around that, instead (this generates tensile forces along the blade and at the blade root, but they can be ignored for the purpose of the discussion).
To get the blade to fly to the new position required by the pilot, inputs are made to change the blade pitch angle. The resulting changes in airflow then alter the aerodynamic force, which overcomes the inertia of the blade. It's position relative to the vertical axis of the mast changes as its inertia is overcome.
As the blade changes its path. it applies forces to the rotor mast, so the helicopter is dragged along behind.....
BTW, gyro theory became more logical to me when I considered what an individual molecule of its mass was doing. It's all about inertia!