Why Does It Come Back?
Unlike an airplane or helicopter propeller, which starts spinning while the vehicle is completely still, you throw the boomerang, so that in addition to its spinning propeller motion, it also has the motion of flying through the air.
In the diagram below, you can see that whichever wing is at the top of the spin at any one time ends up moving in the same direction as the forward motion of the throw, while whichever wing is at the bottom of the spin is moving in the opposite direction of the throw. This means that while the wing at the top is spinning at the same speed as the wing at the bottom, it is actually moving through the air at a higher rate of speed.
When a wing moves through the air more quickly, more air passes under it. This translates into more lift because the wing has to exert more force to push down the increased mass. So, it's as if somebody were constantly pushing the whole spinning propeller of the boomerang at the top of the spin.
But everybody knows that when you push something from the top, say a chair, you tip the thing over and it falls to the ground. Why doesn't this happen when you push on the top of a spinning boomerang?
If you've read How Gyroscopes Work, then you may have already guessed what's going on here. When you push on one point of a spinning object, such as a wheel, airplane propeller or boomerang, the object doesn't react in the way you might expect. When you push a spinning wheel, for example, the wheel reacts to the force as if you pushed it at a point 90 degrees off from where you actually pushed it. To see this, roll a bicycle wheel along next to you and push on it at the top. The wheel will turn to the left or right, as if there were a force acting on the front of the wheel. This is because with a spinning object, the point you push isn't stationary, it's rotating around an axis! You applied the force to a point at the top of the wheel, but that point immediately moved around to the front of the wheel while it was still feeling the force you applied. There's a sort of delayed reaction, and the force actually has the strongest effect on the object about 90 degrees off from where it was first applied.
In this scenario, the wheel would quickly straighten out after turning slightly because as the point of force rotates around the wheel, it ends up applying force on opposite ends of the wheel, which balances out the effect of the force. But constantly pushing on the top of the wheel would keep a steady force acting on the front of the wheel. This force would be stronger than the counterbalancing forces, so the wheel would keep turning, traveling in a circle.
If you've ever steered a bicycle without using the handlebars, you've experienced this effect. You shift your weight on the bicycle so that the top of the wheel moves to the side, but every bicycle rider knows that the bike doesn't tip over as it would if it were standing still, but turns to the right or left instead.
This is the same thing that is happening in a boomerang. The uneven force caused by the difference in speed between the two wings applies a constant force at the top of the spinning boomerang, which is actually felt at the leading side of the spin. So, like a leaning bicycle wheel, the boomerang is constantly turning to the left or right, so that it travels in a circle and comes back to its starting point.