It reduces stray light by only illuminating the part of the specimen which is actually being observed. The higher the magnification of the objective, the smaller the area which can be seen. Köhler illumination limits the light only to the part of the specimen which is currently in the field of view. This increases image contrast because other parts of the specimen are then not able to reflect light from the side into the objective.
Köhler illumination also ensures that the whole specimen is illuminated evenly. Köhler microscopes have an additional optical system and diaphragm mounted right above the lamp. By adjusting the diaphragm, it is possible to limit the light only to the relevant parts of the specimen.