You're witnessing the effects of spin stabilization. It's the same phenomenon that makes rifle bullets fly farther and straighter and why American football players try to throw a spiral.
Your paper airplane has some dynamic instability: there's a positive feedback loop where a small disturbance causes an aerodynamic force that further increases the disturbance. The plane quickly goes out of control and crashes. By introducing rotation, the direction of the force is continuously varied so that the positive feedback loop is broken: the effect of the force summed through a complete rotation cancels out. You get a wobble, but thanks to the rotation it isn't allowed to develop into a full loss of control.
Spin stabilization is also used for spacecraft, but for a different reason: to give the object angular momentum which will dominate over disruptive effects of thermal radiation, gravitational gradients and other nuances of orbital mechanics, allowing one of the orientation axes (the spin axis) to stay more or less fixed. With your paper airplane, the angular momentum isn't very important, but the cancellation of disturbing forces by summing them around a full circle of rotation is.