As heat is applied to the milk, the proteins casein and beta-lactoglobulin start to coagulate, and form a skin on the surface. After further heating, the skin dries because of evaporation, and forms a still firmer barrier. Steam produced under the skin builds up and causes the milk to boil over.
The action of whisking stops the skin from forming (by creating a foam over the surface) and the milk from boiling over. Skin will also form on milky drinks after heating, but adding toppings - marshmallows or whipped cream, say - prevents this.