Why does increase in headwind caused by wind shear increase the speed of an aircraft?

You are forgetting that aircraft have inertia - in a sudden change of wind, the forces acting on the plane will equalize, and the ground speed will change, but that does not occur instantly. Until it does, the airspeed will be effected. Wind shear is actually less of a problem for light aircraft than is is for a larger airliner. If a Cessna 150 is hit with a sudden 15 knot headwind, it may show a 15 kt increase in airspeed for a fraction of a second, but with its low mass it's aerodynamic forces will cause it to decelerate back to the original airspeed very quickly. A 150,000 lb airliner is going to have so much inertia that it will take much longer for it to accelerate or decelerate when hit by a gust. That's also why a smaller plane is "bumpier" to ride in - it also accelerates vertically in response to vertical air currents, where the same vertical currents may not produce a noticeable movement of an airliner

You are thinking too hard. airplanes fly because of airspeed. If you had a rc plane sitting on your desk and you blew air at the front of it with a fan or with compressed air the airspeed would in increase. In reality it will only increase temporarily because windshear happens so fast. just as was already explained so well by the first person.

An suddenly increased headwind or a decreased tailwind results (temporarily) in increased airspeed, lift, and drag. It also results in slower groundspeed.