I have a simple circuit as in the picture below. My question is why the negative charge -Q at the bottom plate of C1 moves and neutralizes with the positive charge +Q at the top plate of C2?
The area inside the green circle is electrically isolated from the rest of the circuit so there is always the same amount of charge inside the circle.
What charge flows into the top of C1, the same amount flows out of the bottom of C1 and into the top of C2, the same amount flows out of the bottom of C2, this is how capacitors work (or at-least a very good approximation)
Why does it move? Electric potential pushes it..
See also: Kirchoff's loop law
If you removed the top plate of the top capacitor and the bottom plate of the bottom capacitor, that's exactly what would happen. However, the presence of those charged plates creates an electric field that keeps the charges separated.
For the negative charge at the top capacitor to flow down, it would have to get away from the positive charge at the top plate of the top capacitor, which attracts it very strongly. Ditto for the bottom capacitor.
In fact, the attraction between the opposite charges at the two plates of a capacitor are exactly what makes it work - otherwise, you wouldn't be able to keep a lot of charge together.
The charges in the green circle will neutralize. It is impossible for them not to neutralize. The forces pulling them apart and the forces pulling them together are the same. The forces balance each other out and they will neutralize.