How Helicopters Change Attitude (Gyroscopic Precession or Thrust Vector Tilt)
This animation shows it better than most.  The swash plate changes the plane of the rotor disc to change the plane of the lift vector.  It's the same thing as an airplane turning by banking.  When an airplane banks it imparts a lateral thrust component to the vertical lift component and it moves sideways, and because it's moving forward as it does so, it moves in a horizontal arc, or a turn.  A helicopter's rotor is just able to "bank" in all directions at any speed.  

The swash plate directly flies the blades to the path that will put the overall disc in the desired plane of rotation.  The blades have to fly to their position using the air; if you had the machine running in a vacuum, the cyclic would have no effect.  However, gyro precession is a factor insofar as the input/result of the swashplate's effect on the blade's physical path is offset 90 degrees.  

That is, to get the blade tip to its highest point in the circle, the highest blade angle, or highest lift force in the rotational cycle, is imparted 90 degrees before that point.  To tilt the rotor disc forward say, the blade that is on its way to its high spot over the tail boom will have reached its highest blade angle while passing at 90 deg on the left.

On the Bell rotor in the animation you can clearly see this.  The rotor disc angle follows the swash plate angle directly - swash plate tilts forward, rotor tilts forward. However, the linkage being driven by the swash plate acts on blade angle with a 90 degree offset (the control rods coming up on each side of the hub) to account for precession.

The Bell teetering system is interesting because it imparts no bending forces whatsoever to the rotor mast. The machine's body is suspended below the rotor hub as if it was a tennis ball on a string.  When you move the stick you are making the rotor disc fly around on its own account by tilting this way and that and the machine suspended under it follows along. In fact if a bending moment WAS applied to the mast, by having the teetering stops contact the mast while it's spinning, it comes apart and you become a falling object (this is "mast bumping" which cause the deaths of a number of Huey pilots in the 60s and Robinson pilots in the 90s). 

Rigid and articulated rotor systems do impart some rolling and pitch moments into the air frame and the rotor head, mast and suspension system for the transmission are designed to transfer these loads.

A good fixed wing analogy between an articulated and teetering rotor is to compare regular airplanes vs hang gliders and trikes.  An articulated/semi rigid rotor is a bit like a regular aircraft where the body is directly acted on by the wings, although with a flexible connection between the wings and fuselage.  The Bell system is more like a hang glider or trike where the wing and body are on a fully flexible connection and you basically fly the wing around and the body underneath follows along without ever reaching the physical limit of the connection.  It's not a perfect analogy but sufficient for visualization.