Wing tip vortices are created by span wise air flow from the lower wing surface (area of higher pressure) to the upper wing surface (area of lower pressure). Generally, the airflow on the upper surface is forced towards the fuselage, whilst the airflow on the lower surface is being forced toward the tip. This results in the airflow from the lower side trying to climb over the wing tip and move to the area of lower pressure, the upper surface.
The winglets act like a solid fence, stopping this movement of air from the lower to upper surface. It doesn’t stop these vortices completely, as the winglet is essentially just an extension to the wing, but does reduce them significantly, as shown in the diagrams above.