Why are the European Ariane rockets launched from the Korou site in French-Guyana, South-America, and not simply from some place in Europe where the rockets are partly built?

The nearer the site for launching rockets into space is to the equator, the better. That is because then, due to the rotation of the earth, the rockets can get an enormous push of about 40.000 / 24 = 1700 km/h. That saves fuel and hence money.

French-Guyana is close to the equator and is besides French teritory, so therefore the European rockets are launched there. For the same reason, the Americans launch their rockets from Florida, which is their most southern state.