The answer is the wave is not actually changing. Think of it this way. When a car playing loud music drives by, the pitch increases as it approaches, sounds correct for a moment, then the pitch decreases as it moves away. The person in the car hears the same pitch the whole time.
Additionally, imagine a car is driving 40 mph toward a wall which is moving 0 mph. The collision will occur with results consistent with a 40 mph collision. Now if a car is driving 40 mph toward another car which is driving 40 mph toward it (head on), a collision will occur with results consistent with a car driving 80 mph toward a wall which is moving 0 mph. If a car which is driving 40 mph toward another car which is driving 40 mph from behind (rear end), a collision will not even occur...so let us imagine the car in the rear is driving 41 mph...a collision will occur which is consistent with a car driving 1 mph toward a wall which is moving 0 mph.
In other words:
The music which has exited the car through the window and has been heard by someone standing outside the car, is the sound which is experienced as being different.
The music is a higher pitch while the car is moving toward the person because the speed* of the wave is increased by the speed of the car. The pitch sounds normal for a moment because the car is not moving toward or away from the person which leaves the speed* of the sound as it is, and the pitch is lower as the car is moving away because the speed* of the sound wave is reduced by the speed of the car.
Application to the microwave scenario:
The microwave oven is like the car and the electromagnetic wave (visible or microwave) is like the sound wave of the music.
The electromagnetic wave being shifted is the one which has left the interior of the microwave oven.
The shift being observed in the electromagnetic wave is caused by a change in the speed* of the wave as it is observed relative to the observer.
Edit: *speed of the waves therefore increased frequency.