For the original poster, the other part of the puzzle which it seems is being left out in some explanations above is drag.
A good and simple way of looking at it is this:
Imagine the spinning rotor as a disc, free to fly where it wants, and the body as a box hanging freely by the mast from the rotor head. As the disc tilts and flies off in any direction (let's say forward with respect to the body), it pulls the mast head along with it. As speed increases, drag builds up on the body trying to push it backwards and therefore aligning it into its proper orientation with the rotor disc - we have a force couple, the forward pull at the mast head vs the backward pull of drag working lower down.
Why doesn't the body just keep moving back? Because, as has been mentioned above, its center of gravity (which is lower than the mast head) is moving backwards from under its suspension point, and the further it goes, the greater the restoring force (an opposing force couple created by the upward pull on the mast vs the downward pull at the centre of gravity).
When the two force couples are equal and opposite, we get a steady state. The faster you go the greater the pull/drag couple becomes, so the body swings further back until the lift/weight couple increases enough to balance it.
That hopefully makes it clear, as long as you're OK with the concept of a force couple (two opposing forces separated by an arm, which creates a rotation - like two hands on a steering wheel, one pushing up and the other down - the bigger the wheel / longer the arm, the greater the twisting force). Apologies if you already knew all that!